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area of a triangle in terms of the median is expressed as follows:

The area of a triangle can be expressed in terms of the medians by:



where:

.

see http://mathworld.wolfram.com/TriangleMedian.html for details.

in terms of your problem:

m1 = 4
m2 = 5
m3 = 6


area of the triangle is therefore:



this becomes:



which becomes:



which becomes:



which becomes

A = 13.22875656

here's another reference:

http://www.math10.com/en/geometry/median.html

here's yet another.

http://pballew.net/medians.htm

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assume his salary starts at 1.
at the beginning of the first year he is making 1.
at the end of the first year he is making 1 * 1.1 = 1.1
at the end of the second year he is making 1.1 * 1.2 = 1.32
at the end of the third year he is making 1.32 * 1.25 = 1.65

he starts at 1 and at the end of the third year he is making 1.65

his average annual increase is given by the compounding formula of:

1.65 = 1*(1+x)^3

this is equivalent to:

1.65 = (1+x)^3

take the cube root of both sides to get:

(1.65)^(1/3) = 1+x

subtract 1 from both sides to get:

x = (1.65)^(1/3) - 1

solve for x to get:

x = .18166575

at the beginning of the first year he is making 1.
at the end of the first year he is making 1 * 1.18166575 = 1.18166575
at the end of the second year he is making 1.18166575 * 1.18166575 = 1.396333946
at the endof the third year he is making 1.396333946 * 1.18166575 = 165

1 * (1.18166575)^3 = 1.65

the equation that was used was the future value of a present amount formula that is equal to:

f = p * (1+i)^n

f = future value
p = present amount
i = interest rate per time period
n = number of time periods.

to solve this problem, we first had to find f.

that was done using the year by year analysis up top.

once we knew f, we could then substitute in the formula to get:

f = 1.65
p = 1
i = x
n = 3

we then solved for x.

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mixture A is 7.5% acid.

let x = amount of mixture A.

7.5% of x is equivalent to .075 * x (you have to divide percent by 100% to get proportion).

this means that .075 * x = amount of acid in mixture A.

let y = amount of mixture C.

then 1.0 * y = amount of acid in mixture B.

this is because 100% of mixture B is equivalent to 1.0 * mixture B.

mixture C is equal to 200 liters.

this means that:

x + y = 200

mixture C will contain 10% acid which means that mixture C will contain .10 * 200 = 20 liters of acid.

this means that:

.075 * x + y = 20

you have 2 equations that need to be solved simultaneously.

they are:

x + y = 200 (first equation)

.075*x + y = 20 (second equation)

you can solve for y in either equation and then solve for x in the other equation.

we'll solve for y in the first equation to get:

y = 200-x

we'll substitute for y in the second equation to get:

.075*x + (200-x) = 20

remove parentheses to get:

.075*x + 200 - x = 20

combine like terms to get:

-.925*x + 200 = 20

subtract 200 from both sides to get:

-.925*x = 20-100 = -180

divide both sides by -.925 to get:

x = -180 / -.925 = 194.5945946

use this value of x to solve for y in the first equation to get:

y = 200 - 194.5945946 = 5.405405405

use the values for x and y in the second equation to confirm that they are good.

the second equation is:

.075*x + y = 20

substituting for x and y in that equation, we get:

.075*194.5945946 + 5.405405405 = 20

this becomes 20 = 20 which is true confirming our values for x and y are good.

your answer is:

you need to add 5.405405405 liters of pure acid to make a 200 liter mixture of 10% acid.

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your original equation is:

w^2+3w-10=0

looks like it should factor to be:

(w+5) * (w-2) = 0

multiply (w+5) * (w-2) and you get:

w^2 - 2w + 5w - 10

combine like terms to get:

w^2 + 3w - 10 which is the same as your original equation, confirming that the factors are good.

your equation becomes:

(w+5)*(w-2) = 0

this means that:

either w+5 = 0 or w-2 = 0 or both.

this means that:

w = -5 or w = 2

substitute in your original equation to confirm.

your original equation is:

w^2+3w-10=0

when w = -5, this equation becomes 25 - 15 - 10 = 0

when w = 2, this equation becomes 4 + 6 - 10 = 0

both equations are true confirming the values for x are both good.

your answer is:

the roots of this equation are:

w = -5 and w = 2

a graph of your original equation is shown below confirming graphically that these roots are accurate.

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formula for volume of a sphere is:



formula for surface area of a sphere is:



if we solve for r in the equation for the volume of the sphere, we get:



take the cube root of both sides of this equation to get:



if substitute for the value of r in the equation for the surface area of the sphere, then we get:

becomes:



if we double v, then this equation becomes:



if we divide s[2] by s[1] we get:



this simplifies to:

which is the same as:



multiply both sides of this equation by and you get:



for example:

let the volume of the sphere be equal to 300.

we solve for the radius to get:

radius of the sphere equals 4.152830592

we confirm by substituting in the equation for the volume of the sphere to get:

v = 4/3 * pi * (4.152830592)^3 = 300 cubic units.

the surface area of the sphere is equal to 4 * pi * r^2 = 216.7196518 square units.

if the sphere doubles in volume, then the surface area of the new sphere should be equal to 216.7196518 * making the surface area of the new sphere equal to

square units

to confirm this is correct, we need to solve for the radius of the new sphere.

the volume of the new sphere is equal to 600 cubic units.

the radius of the new sphere is calculated to be 5.23223868 units.

the surface area of the new sphere is equal to 4 * pi * r^2 which becomes



since this is the same as then we're good.

the answer to your question is:

the surface area was multiplied by .

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if the diameter of the can is increased by 30%, then the radius of the can is also increased by 30%.

consider.

d2 = d1 * 1.3 = 1.3*d1
d1 = r1*2
d2 = 1.3*d1 = 1.3*2*r1
r2 = d2/2 = 1.3*2*r1/2 = 1.3*r1

volume of the can is equal to pi*r^2*h

increase the radius by 30% and you get volume of the can is equal to pi*(1.3*r)^2*h.

in order for the volume of the enlarged radius can to equal 3 times the volume of the original can, the following equation needs to be satisfied.

let x = the amount the height of the can has to be increased.

then:

let v[1] = the volume of the original can.

let v[2] = the volume of the can with a 30% increase in the length of the radius and an unknown increase in the height of the can.

let x = the unknown increase in the height of the can.

v[1] = pi*r^2*h

v[2] = pi*(1.3*r)^2*x*h = 3 * v[1] = 3 * pi*r^2*h

this results in:

pi * (1.3*r)^2 * x * h = 3 * pi * r^2 * h

you want to solve for x.

divide both sides of this equation by h to get:

pi * (1.3*r)^2 * x = 3 * pi * r^2

divide both sides of this equation by (pi * (1.3*r)^2 to get:

x = 3 * pi * r^2 / (pi * (1.3*r)^2)

(1.3*r)^2 becomes (1.3)^2 * r^2. substitute in equation to get:

x = 3 * pi * r^2 / (pi * (1.3)^2 * r^2)

pi and r^2 cancel out from numerator and denominator to get:

x = 3 / (1.3)^2 = 1.775147929

to make the equations equivalent, the height needs to be multiplied by 1.775147929 which is the same as (3/(1.3)^2).

substitute in the original equation to confirm that this is true.

the original equation is:

pi*(1.3*r)^2*x*h = 3 * pi*r^2*h

replace x with (3/(1.3)^2) to get:

pi*(1.3*r)^2*(3/(1.3)^2)*h = 3 * pi*r^2*h

simplify by removing parentheses to get:

pi * (1.3)^2 * r^2 * 3 * h / (1.3)^2 = 3 * pi * r^2 * h

(1.3)^2 in numerator and denominator cancel out to get:

pi * r^2 * 3 * h = 3 * pi * r^2 * h

except for the order in which the terms are presented, the expressions on each side of the equal side are identical confirming that the value of x = (3/(1.3)^2) is correct.

your answer is:

the height needs to be multiplied by (3/(1.3)^2) = 1.775147929 in order for the volume of the resulting can to be tripled, assuming that the diameter has been increased by 30%.

example:

let r = 15
let h = 22

volume of the original can is pi*r^2*h = pi*(15)^2*22 = 1550.88364

multiply radius by 1.3 to get r = 19.5
multiply height by (3/(1.3)^2) to get h = 39.05325444

volume of the enlarged can is pi*r^2*h = pi*(19.5)^2*39.05325444 = 46652.65091.

46652.65091 / 1550.88364 = 3 confirming the value of x is good.

the answer to the question is that the height should be increased by approximately 77.51%.

the original height is x.

the new height is 1.775147929 times x

new height minus old height times 100% equals the percent increase.

1.775147929 * x - x = .775147929 * x * 100% equals 77.5147929% * x = ~ 77.51% * x.

~ means approximately after rounding to the nearest hundredth of a percent.

the height needs to be increased by approximately 77.51%.

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(-6/y)^3 = (-6)^3/y^3 = -126/y^3

this is because (a/b)^3 = a^3/b^3

let a = 5 and let b = 9

(a/b)^3 = (5/9)^3 = .171467764 (use your calculator to solve)

(a/b)^3 = a^3/b^3 = 5^3/9^3 = 125/729 = .171467764

they are equivalent.

what you were asking me to solve is not an equation, but an expression.

(-6/y)^3 is an expression.

when you make it equal to something, then the whole thing becomes an equation.

example:

(-6/y)^3 = 99635

now you have an equation.

the expression on the left side of the equal sign equals the expression on the right side of the equal sign.

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looks like you got the factors ok.

your original equation is:



you factored correctly to get:



you canceled correctly to get:



it looks like where you went wrong was when you inverted the denominator.

since = , your answer should have been:

which becomes:

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formula is:

f = p * (1+i)^n

f = future amount
p = present amount
i = interest rate per time period
n = number of time periods

interest rate = interest rate percent divided by 100 percent.

example:

3% / 100% = .03 which is the interest rate. 3% was the interest rate percent.

you have to adjust your annual interest rate to an interest rate per time period.

annual interest rate is interest rate per year.

if your time period is in years, no adjustment needs to be made.

if your time period is in months, then the annual interest rate needs to be divided by 12 to become a monthly interest rate.

your number of time periods have to be adjusted if they are not in the same denomination.

if your number of time periods are in years, then your number of years does not need to be adjusted.

if your number of time periods is in months, then your number of years needs to be multiplied by 12 in order to be in months.

------------------------------------------------------------------------------

PROBLEM NUMBER 1

a $1000 investment at a rate of 3% compounded monthly and an $1100 investment at a rate of 3.02% compounded annually.

$1000 at 3% compounded monthly formula would be:

f = p*(1+i)^n
p = 1000
i = 3% / 100% / 12 = .0025
y = number of years
n = y*12

formula becomes:

f = 1000*(1.0025)^(12*y)

$1100 at 3.02% compounded annually formula would be:

p = 1100
i = 3.02% / 100% = .0302
y = number of years
n = y

formula becomes:

f = 1100*(1.0302)^(y)

for the investments to be equal, f[1] = f[2] making the equations equal to each other.

you get:

1000*(1.0025)^(12*y) = 1100*(1.0302)^(y)

you need to solve for y.

divide both sides of this equation by (1.0302)^(y) and divide both sides of this equation by 1000 to get:

(1.0025)^(12*y) / (1.0302)^(y) = 1100/1000

take the log of both sides of this equation to get:



since log(a/b) = log(a) - log(b), this equation becomes:



since log(a^b) = b*log(a), this equation becomes:



factor out the y to get:



divide both sides of this equation by to get:



solve for y to get:

y = 454.7149479 years.

the investments will break even in 454.7149479 years.

to confirm this answer is correct, substitute in the original equations.

f = 1000*(1.0025)^(12*y) results in f = 826061905.3

f = f = 1100*(1.0302)^(y) results in f = 826061905.3

they are the same which satisfies the problem statement.

if you take any number of years less than 454, the monthly equation will yield less. if you take any number of years greater than 455, the monthly equation will yield more.

------------------------------------------------------------------------------

PROBLEM NUMBER 2

the continuous compounding formula is different.

that formula would be:



f = future value
p = present amount
r = annual interest rate
y = number of years

you have two formulas that will be compare with each other to provide the solution to this problem.

they are the continuous compounding formula just described, and



f = future value
p = present amount
r = annual interest rate
c = number of compounding periods per year
y = number of years

your problem statement is:

a $2000 investment at a rate of 4% compounded continuously and a $3000 investment at a rate of 5% compounded monthly

with an investment of 2000 at 4% compounded continuously, the formula of:

becomes:



with an investment of 3000 at 5% compounded monthly, the formula of:

becomes:

which becomes:



the investments are equal when the 2 equations are equal to each other.

you get:



divide both sides of this equation by 3000 and divide both sides of this equation by to get:



take the log of both sides of this equation to get:



since , your equation becomes:



since , your equation becomes:



factor out the y to get:



divide both sides of the equation by to get:



solve for y to get:

y = -40.97212172 ?????

I did not expect a negative number of years so I graphed the 2 equations to see what was going on.

The following graph shows that these 2 equations do indeed equal each other when the number of years is negative.

The graphs do not intersect at any positive value of the number of years which means the two equations will never be equivalent.

See below the graph for additional comments.



A calculation when the number of years is equal to -40.97212172 shows that the investments are equal at the value of 388.3929522.

The calculations are correct even if they were confusing.

An analysis of the continuous compounding interest rate of .04 compared to the monthly compounding interest rate of (.05/12) yields the following:

We can convert the monthly compounding rate to the equivalent continuous compounding rate to compare them directly.

The formula to create the equivalent continuous compounding rate from the monthly compounding rate is as follows:

The general process is as follows:

THE PROCESS OF CONVERTING FROM DISCRETE COMPOUNDING TO CONTINUOUS COMPOUNDING IN A NUTSHELL
You are given the nominal rate using discrete compounding.
You solve for the effective rate using discrete compounding.
You use the effective rate using discrete compounding to solve for the nominal rate using continuous compounding.

The actual process given the monthly compounding rate is as follows:

5% = .05 annual interest rate.

.05/12 = .004166667 monthly interst rate.

1.004166667^12 = Effective annual rate using monthly compounding = 1.051161898

Use this effective rate to solve for the continuous compounding rate.

The formula is:

R[e] = e^r which becomes:

1.051161898 = e^r

Take natural log of both sides to get:

ln(1.051161898 = ln(e^r)

since ln(e^r) = r*ln(e) = r*1, this equation becomes:

ln(1.051161898 = r which becomes:

r = .049896122

what this means is that compounding 5% a year monthly is equivalent to continuous compounding at 4.9896122% a year.

For example, consider an investment of 1 at 5% a years with monthly compounding for 5 years and compare it to an investment of 1 at 4.9896122% a year with continuous compounding.

1 * (1.004166667)^(5*12) = 1.283358679
1 * e^(.049896122*5) = 1.283358679

They are equivalent.

The bottom line is:

3000 at 5% a year with monthly compounding equals an equivalent continuous compounding rate of 4.98...% a year.

2000 at a continuous compounding rate of 4% a year will never catch up to it.

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equation is:



* means multiplication

That is what I think your equation is.

rules of exponents you will use:

(rule 1)

(rule 2)

(rule 3)

your equation of becomes:

by rule 1, which becomes:

by rule 2, which becomes:

by rule 3, which becomes:



a common mistake is to take something like and only work on and forgetting to work on as well.

making this mistake, you would get .

not making this mistake, you woul get .

that exterior exponent to the expression has to be applied to every term in the expression, including the coefficient of a term.

in the expression , is the coefficient of the term . It is another term in the expression, just like is.

to prove your answer is correct, take any value for a and b and substitute for them in the original and final equation to see that you got the same answer.

let a = 3
let b = 6

use your calculator to solve.

original equation is:

which becomes:

which becomes:

.000000265

final equation is:

which becomes

which becomes:

.000000265

the answers are the same so the original equation has been simplified correctly.

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what is the exponential equation that you started with?

i didn't see one.

y = 3*(x+4) is not exponential.

is that your original equation?

is your original equation something else?

y = 3*(x+4) is the equation of a straight line.

that would look like this:



an exponential equation would be in the form of:

y = b^x where b is the base and x is the exponent that the base is being raised to.

an example would be:

y = 3^(x+3) where 3 is the base and x+3 is the exponent.

use the ^ symbol to indicate exponentiation.

use parentheses to keep the exponent together if it is an expression with more than 1 term in it.

y^x is ok.

y^x+1 is not ok. it should be written as y^(x+1) to make certain that the whole expression is the exponent.

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log(x-1) + log(5x) = 2

three basic rules of logarithms are:

log(a*b) = log(a) + log(b) (rule 1)
log(a/b) = log(a) - log(b) (rule 2)
log(a^b) = b*log(a) (rule 3)

your equation can be made into the following by invoking rule 1.

log(x-1) + log(5x) = 2 becomes:

log((x-1)*(5x)) = 2

this becomes:

log(5x^2 - 5x) = 2

the basic rule of logarithms states:

y = log(a,x) if and only if a^y = x

in your equation, a base of 10 is implied.

your equation of log(5x^2 - 5x) = 2 is really:

log(10,(5x^2-5x)) = 2 which means:

log of (5x^2-5x) to the base of 10 = 2

using the basic rule of logarithms, this equation becomes:

log(10,(5x^2-5x)) = 2 if and only if:

10^2 = 5x^2-5x

this becomes:

100 = 5x^2 - 5x

subtract 100 from both sides of this equation to get:

5x^2 - 5x - 100 = 0 which is a quadratic equation.

divide both sides of this equation by 5 to get:

x^2 - x - 20 = 0

this factors out to be:

(x-5)*(x+4) = 0

solve for x to get:

x = 5 or x = -4

substitute in original quadratic equation to confirm these answers are good.

5x^2 - 5x = 100 becomes:

80 + 20 = 100 when x = -4 and becomes:

125 - 25 = 100 when x = 5.

both solutions are good.

plug these solutions into your original equation that you started with to get:

log(x-1) + log(5x) = 2 becomes:

log(4) + log(25) = 2 when x = 5.

solving this equations gets 2 = 2 confirming x = 5 is a good solution.

log(x-1) + log(5x) = 2 becomes:

log(-5) + log(-20) = 2 when x = -4

this solutions is not valid because you can't take the log of a negative number.

your only valid solution is:

x = 5

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2 minutes 23 seconds is equivalent to 143 seconds.

2 minutes 15 seconds is equivalent to 135 seconds.

she lowered her time by 8 seconds (135 + 8 = 143).

the percent decrease equals the difference divided by the first time, with the result of that operation being multiplied by 100%. This becomes:

8/143 = .05594405594 * 100% = 5.594405594 percent.

that's pretty close to selection B of 5.6%

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volume = height * area of base.

let v = volume, h = height, a = area of base.

v = h*a

solve for a to get:

a = v/h

v = 48
h = 2

a = 48/2 = 24

the area of the base is 24 square inches.

24 = 1*24 = 2*12 = 3*8 = 4*6

possible dimensions for the base where the dimensions are in whole numbers are:

                   length          width
                     24              1
                     12              2
                      8              3
                      6              4

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formula is:

F = P*e^(kt) where:

F = future amount
P = present amount
k = constant of proportion.
t = time

half life of carbon is given by the equation:

.5 = e^(5600k) because:

half life of carbon is 5600 years.

solve for k and then you can solve the problem.

take log of both sides of equation of .5 = e^(5600k) to get:

log(.5) = log(e^(5600k))

this becomes:

log(.5) = 5600k * log(e)

divide both sides of this equation by 5600*log(e) to get:

k = log(.5)/(5600*log(e))

solve for k to get:

k = -.301029996 / (5600*(.434294482) which becomes:

k = -.000123776

substitute in your original equation to confirm this value for k is good.

.5 = e^(5600*k) becomes:

.5 = e^(5600*-.000123776) which becomes:

.5 = .5 confirming the value for k is good.

now that you have the value of k, you can solve the equation.

if the tree has only 30% of the carbon left, then your equation of:

F = P * e^(kt) becomes:

.3 = 1*e^(k*t) which becomes:

.3 = e^(-.000123776*t)

take the log of both sides of this equation to get:

log(.3) = log(e^(-.000123776*t)) which becomes:

log(.3) = -.000123776*t*log(e)

divide both sides of this equation by -.000123776*log(e) to get:

t = log(.3) / (-.000123776*log(e))

solve for t to get:

t = 9727.007327 years

to confirm this is correct, substitute in the original equation to get:

.3 = e^(-.000123776 * 9727.007327) which becomes:

.3 = .3

the key is knowing the formula f = p*e^(kt) which I hope they gave you.

without the use of the scientific constant of e, you would have had to at least know that the rate of growth or decay is exponential.

solving this without the use of the formula f = p*e^(kt) is possible using exponential growth formula of:

f = p*(1+x)^t

this is a compounding formula where x is the rate of growth or decay.

using this formula, you would get:

f = p * (1+x)^t becomes:

.5 = 1 * (1+x)^5600 which becomes:

.5 = (1+x)^5600

take the 5600th root of both sides of this equation to get:

(1+x) = .5^(1/5600) which becomes:

(1+x) = .999876231

now that you know what (1+x) is equal to, you can solve the equation.

equation of:

f = p*(1+x)^t becomes:

.3 = 1*(.999876231)^t which becomes;

.3 = (.999876231)^t.

take log of both sides of this equation to get:

log(.3) = log(.999876231^t) which becomes:

log(.3) = t*log(.999876231)

divide both sides of this equation by log(.999876231) to get:

t = log(.3) / log(.999876231)

solve for t to get:

t = 9727.007361 years.

that's pretty close to t = 9727.007327 years which is what we got using the exponential formula of f = p*e^(kt)

bottom line is:

your answer is 9727.007327 years assuming exponential rate of growth or decay.

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let:

z = distance zeno has to travel
o = distance that orb has to travel
y = distance that yurko has to travel
s = distance that sam has to travel.

from the problem statement:

o = z/2
y = z + o which is equivalent to z + z/2
s = 3z

total distance they trafel is 888 space miles.

this means that:

z + o + y + s = 888

substituting all values in terms of z, we get:

z + (z/2) + (z + z/2) + 3z = 888

simplifying by removing parentheses gets:

z + z/2 + z + z/2 + 3z = 888

combining like terms gets:

6z = 888

dividing both sides of the equation by 6 gets:

z = 148

o = z/2 = 148/2 = 74

y = z + o = 148 + 74 = 222

s = 3z = 444

148 + 74 + 222 + 444 = 888 confirming the answers are good.

your answer is:

zeno travels 148 space miles
orb travels 74 space miles
yurko travels 222 space miles
sam travels 444 space miles

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let y = one angle.
let x = the other angle.

your equation becomes:

y = 3*x - 4

this means that y is 4 degrees less than 3 * x.

you don't have enough information to solve for the measure of the angle.

you have to be given the value of one or the other in order to solve that.

you do have enough information to solve for the measure of the angle in terms of the measure of the other angle.

using this formula, you can find the value of y given the value of x, but you have to be given the value of x.

if you are not given the value of x, you can't find the value of y, because the value of y will vary based on the value of x.

a graph of your equation shows the relationship.



from this graph, you can see that the value of y is determined by the value of x.

the graph is not accurate enough for you to see this directly, but if you use the formula to solve the equation for a given value of x, you will see that the value of y is in the general vicinity of that answer.

for example, when x = 63, y = 3*63 - 4 = 185 degrees.

I put a horizontal line at y = 185 to show you that the intersection of that line with the graph of the equation occurs at x = 63. drop a vertical line from that intersection and it will intersect the x-axis at approximately x = 63 if not right on. any deviation is due to the inaccuracy of the graphing method and or the display of the graph.

the answer to your question is:

the measure of the smaller angle is x.
the measure of the larger angle is y.

this was determined by relating the 2 angles through the use of a formula that was derived from the given statements about the relationship between the 2 angles.

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the measurement of your yard is:

length 15 yards.
width = 6 years.

the area of your yard is 15 * 6 = 90 square yards.

you want to put a stone border around this yard.

you have 100 square yards of stone available to you.

once you ring your yard with the stone border, the total area will be:

90 square yards + 100 square yards = 190 square yards.

the border will be the same width all around, so we let the width of the border be equal to x.

this means that the length of the garden will be increased by 2*x, and the width of the garden will be increased by 2*x (x added to each end equals a total of 2*x).

our equation for the area of the garden plus the border becomes:

(2x + 6) * (2x + 15) = 190

we multiply these factors together to get:

4x^2 + 30x + 12x + 90 = 190

we subtract 190 from both sides of this equation and we simplify by combining like terms to get:

4x^2 + 42x - 100 = 0

we use the quadratic formula to solve this equation to get:

x = 2 or x = -12.5

x has to be positive, so we take x = 2 and substitute in the original equation to confirm that the equation is true meaning that the value of x is good.

original equation is:

(2x + 6) * (2x + 15) = 190

replace x with 2 to get:

(2*2 + 6) * (2*2 + 15) = 190

simplify to get:

10 * 19 = 190

since this equation is true, the value of x = 2 is good.

the answer to the problem is:

the width of our border will be 2 yards.

USE OF QUADRATIC FORMULA

the quadratic formula is:



the standard form of a quadratic equation is:

ax^2 + bx + c = 0

your formula was:

4x^2 + 42x - 100 = 0

this meant that:

a = 4
b = 42
c = -100

using these values, the quadratic formula became:



simplifying made the equation look like:



simplifying further made the equation look like:



simplifying further made the equation look like:



the rest was just finishing up work by performing the indicated operations.

FIVE STEP PROCESS

We followed the 5 step process even though you may not have realized it.

The 5 step process is as follows:

1. REad the problem
2. Choose a variable to represent one quantity. Use that variable to represent every quantity in the problem.
3. Write an equation relating the quantities in the problem.
4. Solve your equation and answer the question asked.
5. Check your answer.

We read the problem. (step 1)
We chose the variable x to represent the width of the border (step 2)
We wrote an equation relating the quantities in the problem. That was the equation of (2x+6) * (2x+15) = 190 (step 3)
WE solved the equation and answered the question asked (step 4).
We checked our answer to confirm that it was good (step 5).

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the slope for y = 3 is 0.

the slope for x = -2 is undefined.

they are perpendicular to each other because the slope of 1 is a negative reciprocal of the other.

negative reciprocal of a slope is equal to -1 divided by the slope.

negative reciprocal of 0 is equal to -1/0 which is undefined.

since the slopes of the 2 lines follow the rules for being negative reciprocals of each other, then the lines are perpendicular to each other.

standard form of the equation of a line is ax + by = c

y = 3 converted to this form is equivalent to:

0*x + 1*y = 3

x = -2 converted to this form is equivalent to:

1*x + 0*y = -2

since the terms being multiplied by 0 cancel out, these equations become:

y = 3 and x = -2

the equations you are seeing are in standard form.

the slope intercept form of these equations is:

y = m*x + b where:

m is the slope.
b is the y-intercept.

we will convert the standard form of these equations to slope intercept form so show you what is happening with the slope.

we will take the original standard form before the canceling out of the terms.

they are:

0*x + 1*y = 3

1*x + 0*y = -2

to convert these, you solve for y in both equations.

the first equation is:

0*x + 1*y = 3

subtract 0*x from both sides of the equation to get:

1*y = -0*x + 3

divide both sides of this equation by 1 to get:

y = (-0/1)*x + (3/1)

this is in slope intercept form with m*x = (-0/1)*x, so the slope is equal to (-0/1) which is equivalent to 0.

y = 3 is equivalent to standard form of 0*x + 1*y = 3 which is equivalent to slope intercept form of y = 0*x + 3 with a slope of 0.

the second equation is:

1*x + 0*y = -2

subtract 1*x from both sides of this equation to get:

0*y = -1*x - 2

divide both sides of this equation by 0 to get:

y = (-1/0)*x - (2/0)

this is in slope intercept form with m*x = (-1/0)*x, so the slope is equal to (-1/0) which is undefined.

x = -2 is equivalent to standard form of 1*x + 0*y = -2 which is equivalent to slope intercept form of y = (-1/0)*x - (2/0) with a slope that is undefined.

note that the y-intercept is equal to (2/0) which is also undefined.

a slope of (-1/0) is the negative reciprocal of a slope of 0 meaning that y = 3 and x = -2 are perpendicular to each other.

y = 3 is a horizontal line 3 units above the x-axis.

x = -2 is a vertical line 2 units to the left of the y-axis.

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i = measure of the interior angle of regular polygon.
e = measure of the exterior angle of regular polygon.

i = 6*e + 12

s = number of sides in regular polygon.

exterior angle = 360 divided by number of sides = 360/s

interior angle = (s-2)*180 / s

example:

exterior angle of regular triangle = 360 / 3 = 120

interior angle of regular triangle = 1*180/3 = 60

the interior angle of a polygon and its associated exterior angle are supplementary to each other. the sum of their angles is 180.

we get i + e = 180

we also get:

i = 6*e + 12

substitute in supplementary equation to get:

6*e + 12 + e = 180

combine like terms to get:

7*e + 12 = 180

subtract 12 from both sides of equation to get:

7*e = 180 - 12 = 168

divide both sides of equation by 7 to get:

e = 168/7 = 24.

i = 6*24 + 12 = 156

156 + 24 = 180 so the angles are supplementary as they should be.

we have:

i = 156
e = 24

formula for exterior angle of a regular polygon is:

e = 360/s

this becomes

24 = 360/s

solve for s to get:

s = 15

number of sides in the regular polygon appears to be 15.

substitute in equation for interior angle of polygon to get:

156 = (s-2)*180/s

solve for s.

mulltiply both sides of the equation by s to get:

156*s = (s-2)*180

simplify by removing parentheses to get:

156*s = 180*s - 360

solve for s to get:

24*s = 360

s = 360/24 = 15

everything checks out so the answer is:

number of sides of the regular polygon is 15.

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x = number of quarts at 4% butterfat.
y = number of quarts at 1% butterfat.

you want to get 90 quarts of 3% butterfat.

amount of butterfat you get in the mixture will be .03 * 90 = 2.7 quarts of butterfat.

.04 * x + .01 * y = 2.7 (equation 1)

x + y = 90 (equation 2)

equation 1 tells you how much of x and y you need to get 2.7 quarts of butterfat.

equation 2 tells you how much of x and y you need to get 90 quarts of milk total.

you need to solve these equations simultaneously so the same value of x and y satisfies both equations.

solve for x in equation 2 to get:

x = 90 - y

substitute for x in equation 1 and solve for y.

.04 * x + .01 * y = 2.7 (equation 1) becomes:

.04 * (90-y) + .01*y = 2.7

simplify by removing parentheses to get:

.04*90 - .04*y + .01*y = 2.7

simlify by performing indicated operations and combining like terms to get

3.6 - .03y = 2.7

add .03y to both sides of the equation and subtract 2.7 from both sides of the equation to get:

3.6 - 2.7 = .03y

combine like terms to get:

.9 = .03y

divide both sides of the equation by .03 to get

y = 30

if y = 30, then x = 90-y = 60
x = 60
y = 30

substitute in original equation to confirm the answer is true.

original equation is:

.04 * x + .01 * y = 2.7

substitute for x and y to get:

.04 * 60 + .01 * 30 = 2.7

this becomes 2.7 = 2.7 which is true, confirming that the answer is correct.

the answer is:

the mixture should contain 60 quarts of 4% butterfat milk.
the mixture should contain 30 quarts of 1% butterfat milk.

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in the following equations:

y = total cost
m = cost per movie
x = number of movies per year
f = fixed cost per year

general equation is:

y = m*x + f

for members:

m = $2.00 per movie
f = $75.00 per year

member equation is:

y = 2*x + 75

for non-members:

m = $5.75

non-member equation is:

y = 5.75*x

break even point is when the 2 equations are equal to each other.

2*x + 75 = 5.75*x

subtract 2*x from both sides of the equation to get:

3.75*x = 75

x = 75/3.75 = 20

At 20 movies in a year, they break even. above 20 movies in a year, the total cost to the member becomes less than the total cost to the non-member.

member equation is:

y = 2x + 75
when x = 20, y = 40+75 = 115

non-member equation is:

y = 5.75*x
when x = 20, y = 20*5.75 = 115

anything over 20, the member pays less.

assume x = 30

member pays 2*30 + 75 = 60 + 75 = 135
non-member pays 5.75*30 = 172.5

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x + y = 5000
x * .12 + y * .10 = 568

use first equation to solve for y in terms of x.

y = 5000 - x

substitute for y in second equation to get:

x * .12 + (5000-x) * .10 = 568

simplify by removing parentheses to get:

x * .12 + 5000*.10 - x*.10 = 568

combine like terms and simplify further to get:

.02*x + 500 = 568

subtract 500 from both sides to get:

.02*x = 68

divide both sides by .02 to get:

x = 68/.02 = 3400.

if x = 3400, then y = 5000 - 3400 = 1600.

.12 * 3400 + .10 * 1600 = 408 + 160 = 568.

his confirms the answer is good.

the answer is:

geoff invests 3400 at 12% and 1600 at 10% to gain 568 interest in one year.

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the rules of logarithms state:

(rule 1)

(rule 2)

(rule 3)

your expression is:



by rule 3.

by rule 3.

your expression becomes:



this becomes by rule 1.

since can also be written as , this can also be written as:



you can prove that your original expression of is the same as your final expression of by substituting for the values of a, y, z.

I used a = 10 because this base is solvable by using the LOG function of the calculator.

I let y = 7
I let z = 15

I got the same answer for the original expression and the final expression which says that I made the translation correctly.

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the graph will be the same as y = 16x once you simplify the equation.

the graph looks like this:



you graph it like you would any other graph.

you pick values of x and you solve for y.

if you pick x = 0, they y = 0

if you pick x = 3, then 16*x = 16*3 = 48

note that y = 6x + 10x becomes y = 6*3 + 10*3 = 18 + 30 = 48 (same answer).

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Use of Heron's formula may help.

That gives you the area of a triangle given 3 sides of the triangle.

The formula is:

Area=sqrt(s(s-a)(s-b)(s-c))

s = sum of the sides of the triangle divided by 2.
a,b,c are the sides of the triangle.

your trapezoid looks like and is labeled like this:

                           A                           B
                           xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
                         x x                           xx
                      x    x                           x x
                   x       x                           x  x
                x          x                           x   x
             x             x                           x    x
          x                x                           x     x
       x                   x                           x      x
     xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
     C                     D                           E       F 

your trapezoid is ABFC
you drop an altitude from A to D and from B to E.
you have a rectangle at ABED
you have 2 triangles at ADC and BEF
CA is equal to 15
AB is equal to 10
BF is equal to 13
CF is equal to 24
DE is equal to 10 because it is the opposite side of AB in rectangle ABED.

the length of your top base is AB = 10
the length of your bottom base is CF = 24 with DE representing 10 of it.

if you remove the rectangle from the diagram, you are left with 2 triangles that connect together and form 1 triangle that has sides of 15, 13, and a base of 14.

your modified diagram looks like this:

                           A                          
                           x
                         x xx
                      x    x x
                   x       x  x
                x          x   x
             x             x    x
          x                x     x
       x                   x      x
     xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
     C                     D       F 

CA = 15
AF = 13
CF = 24 - 10 = 14

the sum of the perimeter is equal to 15 + 13 + 14 = 42

Heron's formula says the area of this triangle is equal to:

Area=sqrt(s(s-a)(s-b)(s-c))

where s = (a+b+c)/2 = 42/2 = 21

Formula becomes:

Area = sqrt (21 * (21-15) * (21-13) * (21-14))

This resolves to:

Area = 84

Now that we have the area of this triangle, we can find the height of the triangle which is also the height of the trapezoid.

The area the triangle is equal to 1/2 * b * h.

Since the area of the triangle is 84, this becomes:

84 = 1/2 * b * h

Since the base of the triangle is 14, this becomes:

84 = 1/2 * 14 * h

solve for h to get:

h = 84 * 2 / 14 = 12

now that we have h we can solve for the area of the trapezoid.

Area of the trapezoid = (1/2) * (b1+b2)*h.

This equals to (1/2) * (10+24) * 12 = (1/2) * (34) * 12

Area of the trapezoid = 204

Here's a link to Heron's formula.

http://mste.illinois.edu/dildine/heron/triarea.html

I don't know if there's another way to solve this. I tried to find a way to get the height but was unsuccessful. It just didn't appear there was enough information provided.

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formula is:

h(x) = 2.75*x + 71.48

replace x with 33 to get:

h(33) = 2.75*33 + 71.48 to get:

h(33) = 162.23

if her humerus is 33 centimeters, then her overall height should be 162.23 centimeters.

your answer is:

the woman's height is 162.3 centimeters.

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the possible combinations of boys and girls are:

bbbb
bbbg
bbgg
bggg
gggg

all boys can occur in 1 way so p(4b) = 1 * .5^4 = .0625
3 boys 1 girl can occur in 4 ways so p(3b1g) = 4 * .5^4 = .25
2 boys 2 girls can occur in 6 ways so p(2b2g) = 6 * .5^4 = .375
1 boy 2 girls can occur in 4 ways so p(1b3g) = 4 * .5^4 = .25
all girls can occur in 1 way so p(4g) = 1 * .5^4 = .0625

sum of all probabilities should be equal to 1.

2 * .0625 + 2 * .25 + .375 = 1 so we're good there.

probability of at least 2 boys equals p(2b2g) + p(3b1g) + p(4b) = .375 + .25 + .0625 = .6875 = 68.75%

the number of ways you can get all boys is 1 as shown below:

bbbb

the number of ways you can get 3 boys and 1 girls is 4 as shown below:

bbbg
bbgb
bgbb
gbbb

the number of ways you can get 2 boys and 1 girl is 6 as shown below:

bbgg
bggb
ggbb
gbbg
bgbg
gbgb

since the order in which the children arrive has something to do with the probability of them arriving, these would be permutations.

since the probability of having a boy or a girl was the same (.5 in each case), the probability part of the equations was always the same (.5^4) regardless of whether it was 1 boy, 2 boys, 3 boys, or 4 boys.

for example:

p(bbbb) = .5*.5*.5*.5 = .5^4
p(gggg) = .5*.5*.5*.5 = .5^4
p(bbbg) = .5*.5*.5*.5 = .5^4
p(bbgg) = .5*.5*.5*.5 = .5^4

if the probability was different, i.e. p(b) = .75, p(g) = .25, then the probability for each of the permutations would have been different.

for example:

p(bbbb) = .25*.25*.25*.25 = .25^4
p(gggg) = .75*.75*.75*.75 = .75^4
p(bbbg) = .25*.25*.25*.75 = .25^3*.75
p(bbgg) = .25*.25*.75*.75 = .25^2*.75^2

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a gon is a grad is a gradian.

there are 400 gradians to a circle.

there are 360 degrees to a circle.

let g = gon = gradian

let d = degree

your conversion formulas are:

g = d/.9

d = g*.9

if g = 400, then d = 400*.9 = 360 degrees

if d = 360, then g = 360/.9 = 400 gradians

you have g = 60 gradians

d = g*.9 = 60*.9 = 54 degrees

60 gradians goes into 400 (6 and 2/3) times.

54 degrees goes into 360 (6 and 2/3) times.

60 gradians is equivalent to 54 degrees.

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N = Ie^(kt)

Let I = 1

If in 1 year it decays by .275%, then at the end of the first year it will be reduced from 1to 1 - .00275 = .99725

your formula becomes:

.99725 = 1 * e^(k*1) which becomes:

.99725 = 1 * e^k

which is the same as:

.99725 = e^k

take the log of both sides to get:

log(.99725) = log(e^k)

this becomes:

log(.99725) = k*log(e)

divide both sides by log(e) to get:

log(.99725) / log(e) = k

solve for k to get:

k = -.002753788

now that you have found k, you can find the half life.

to find the half life, let I = 1 and let N = .5 and substitute in general equation of N = I * e^(kt) to get:

.5 = 1 * e^(-.002753788*t)

this is the same as:

.5 = e^(-.002753788*t)

take the log of both sides to get:

log(.5) = log(e^(-.002753788*t)

this becomes:

log(.5) = -.002753788*t*log(e)

divide both sides of this equation by -.002753788*log(e) to get:

t = log(.5)/(-.002753788*log(e))

solve for t to get:

t = 251.7067876 years

The half life of this radioactive isotope is 251.70657867 years.

plug that value in the original equation to get:

N = 1 * e^(kt) becomes

N = 1 * e^(-.002753788*251.7067876)

Solve for N to get:

N = .5

.5 is one half of 1 so the half life of the isotope is equal to 251.7067876 years.

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C(x) = cost to produce all bicycles.
A(x) = average cost to produce one bicycle.

C(x) = 25*x + 20000

A(x) = (25*x + 20000)/x

C(250) = 25*(250) + 20000 = 26250

A(250) = 26250 / 250 = 105

C(250) = A(250) * 250 = 26250

Total cost to produce 250 bicycles is $26,250.

The average cost to produce each bicycle is $105.