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The population of a city was 127 thousand in 1992. The exponential growth rate was 1.3% per year. Find the exponential growth function in terms of t, where t is the number of years since 1992. P(t)=?
The general exponential growth/decay model is:
A = Pe^(rt)
where
A is the amount after time t
P is the initial amount
r is the rate of growth or decay
t is time
.
Plugging in what was given by the problem:
P(t)= 127e^(.013t)

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They are applying the Pythagorean theorem:
a^2 + b^2 = c^2
where
'a' and 'b' are the legs of a right triangle
and 'c' is the hypotenuse (longest side)
4^2 + 4^2 = c^2
16 + 16 = c^2
32 = c^2
sqrt(32) = c
5.66 = c

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Find the slope,if it exists,of the line containing the pair of points.
(2,10) and (8,-4)
The slope
m=(y2-y1)/(x2-x1)
m=(-4-10)/(8-2)
m=(-14)/(6)
m=-7/3

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If it takes Steve 5 hours to mow the lawn, and it takes Jane 3 hours to mow the same lawn, how long does it take them to mow the lawn together?
.
Let x = time (hours) it takes for both
then
x(1/5 + 1/3) = 1
multiplying both sides by 15:
x(3 + 5) = 15
x(8)=15
x = 1.875 hours
or
x = 1 hour and 53 minutes

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Dal and Kim can assemble a swing set in 1 1/2 hr. Working alone, it takes Kim 4 hr longer than Dal to assemble the swing set. How long would it take Dal, working alone, to assemble the swing set?
.
Let d = time it takes for Dal alone
then
d+4 = time it takes for Kim
.
1.5(1/d + 1/(d+4)) = 1
multiplying both sides by d(d+4):
1.5(d+4 + d) = d(d+4)
1.5(2d+4) = d^2+4d
3d+6 = d^2+4d
6 = d^2+d
0 = d^2+d-6
0 = (d+3)(d-2)
d = {-3, 2}
toss out the negative solution leaving:
d = 2 hours (Dal's time working alone)

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Write the slope-intercept equation for the line that passes through (-5, -8) and is perpendicular to 10x – 6y = -11 Please show all of your work. .
.
Determine slope of:
10x – 6y = -11
By rearranging into "slope-intercept" form:
10x – 6y = -11
10x = 6y -11
10x + 11 = 6y
(10/6)x + 11/6 = y
(5/3)x + 11/6 = y
slope is 5/3
a line perpendicular must have a "negative reciprocal":
(5/3)m = -1
m = -3/5 (slope of new line)
.
using the slope (-3/5) and (-5, -8) plug into the "point-slope" form:
y - y1 = m(x - x1)
y - (-8) = (-3/5)(x - (-5))
y + 8 = (-3/5)(x + 5)
y + 8 = (-3/5)x - 3
y = (-3/5)x - 11 (this is what they're looking for)

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log345 - log35 = log7x
log(345/35) = log7x
345/35 = 7x
(345/35)/7 = x
1.41 = x

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Find the x- and y- intercepts for the graph of the following equation.
5x + 6y = 120
.
x-intercept is found by setting y to 0 and solving for x:
5x + 6y = 120
5x + 6(0) = 120
5x = 120
x = 24
x-intercept is at (24,0)
.
y-intercept is found by setting x to 0 and solving for y:
5x + 6y = 120
5(0) + 6y = 120
6y = 120
y = 20
y-intercept is at (0, 20)

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If is a cost function, how fast is marginal cost changing when q=100 ?
.
Find the derivative of:

c' =
.
set q to 100 and find c':

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By "looking" at:
y = 0.005x^2 -2x +200
we know that it is a parabola that opens upwards (because of the positive coefficient associated with the x^2 term).
So, the vertex would be the lowest point and where it would turn upwards:
x = -b/(2a)
x = -(-2)/(2*0.005)
x = 2/0.010
x = 200 (this is what they're looking for)

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lorarithmic form
9^4=6561

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Because:
y=(5/3)x+(8/5)
is already in "slope-intercept" form of
y=mx+b
where (0,b) is the y-intecept
.
From inspection, we know the y-intecept of
y=(5/3)x+(8/5)
is at
(0, 8/5)

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The value of fraction is 4/7 and its denominator is 15 greater its numerator. Find the fraction.
.
Let n = numerator
then
n+15 = denominator
.
n/(n+15) = 4/7
cross multiplying:
7n = 4(n+15)
7n = 4n+60
3n = 60
n = 20
.
fraction is then:
n/(n+15) = 20/(20+15) = 20/35

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"Christy drove at 50 miles per hour for h hours but arrived g hours late. How fast should Christy have driven to have arrived on time?"
.
Applying the distance formula of d=rt
Distance needed to travel: 50h miles
Time needed to make it in: h-g
.
since:
r = d/t
r = (50h)/(h-g)
.
Therefore, your answer is (relative to h and g):
(50h)/(h-g) mph

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applying "change of base":

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log(2-x)=.05
(2-x)= 10^(.05)
2-x= 1.122
-x= 1.122-2
-x= -0.878
x= 0.878

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For the cost function c=0.4q^2+4q+5 , find the rate of change of c with respect to q when q=2
.
Begin by finding the first derivative of the function:
c=0.4q^2+4q+5
c'=0.8q+4
.
The "rate of change" when q=2:
c'=0.8(2)+4
c'=1.6+4
c'=5.6

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Find the length of the hypotenuse of a right triangle whose legs measure 5 and 7
.
Applying the Pythagorean Theorem:
Let h = hypotenuse
then
h^2 = 5^2 + 7^2
h^2 = 25 + 49
h^2 = 74
h = sqrt(74)
h = 8.6

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Factor completely: 8x^8 - 64x^7 + 16x^6
factor out 8x^6:
(8x^6)(x^2 - 8x + 2)

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Can 3sqrt(3)+sqrt18 be simplified even more?
Yes...
.



factoring out the 3:

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because
y=-4x+8
is in "slope-intercept" form of:
y=mx+b
we can immediately tell from inspection that the slope is:
-4

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Soymeal contains 16% protein. Cornmeal contains 8% protein. How many pounds of each would you need to make 320 pounds of mixture containing 14% protein?
.
Let x = amount (lbs) of soymeal
then
320-x = amount (lbs) of cornmeal
.
.16x + .08(320-x) = .14(320)
Can you finish from here?

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factor: c^2 + 4c + 4
(c+2)(c+2)
Or
(c+2)^2

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leg1: x+2 leg2: x and the hypotenuse is square root of 10. how to find x using the pythagorean theorem?.
.
Pythagorean theorem says:
(leg1)^2 + (leg2)^2 = (hypotenuse)^2
plug in what was given:
(x+2)^2 + x^2 = (sqrt(10))^2
(x+2)^2 + x^2 = 10
(x+2)(x+2) + x^2 = 10
x^2+4x+4 + x^2 = 10
2x^2+4x+4 = 10
2x^2+4x-6 = 0
x^2+2x-3 = 0
factoring:
(x+3)(x-1) = 0
x = {-3,1}
throw out the negative solution (extraneous) leaving:
x = 1

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The cost per megabyte of disk storage in 1981 was $700. In 2010 the cost per megabyte of disk storage was about $.00012. Assuming that the cost per megabyte is decreasing exponentially, find an exponential model of the form
C(t)=Aekt that gives the cost per megabyte C in year t. Use 1981 for t = 0, and round k to the nearest thousandth. According to the model, in what year was the cost per megabyte of storage $.12?
.
C(t)=Ae^(kt)
C(t) is .00012
A is 700
k is what we're looking for
t is 2010-1981=29
.
.00012 = 700e^(k*29)
.00012/700 = e^(k*29)
ln(.00012/700) = k*29
ln(.00012/700)/29 = k
-0.537 = k
.
our general formula:
C(t)=700e^(-0.537t)
Set c(t) to .12 and solve for t:
.12=700e^(-0.537t)
.12/700 = e^(-0.537t)
ln(.12/700) = -0.537t
ln(.12/700)/(-0.537) = t
16.148 = t
.
1981+16.148 = 1997.148
answer: 1997

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Using the formula f(x) = x^2 – 2x + 1, find if it has a maximum or minimum and give that point. Also give x-intercepts.
.
f(x) = x^2 – 2x + 1
Examining the coefficient associated with the x^2 term, we see that it is "positive" -- this means the vertex is a MINIMUM.
x = -b/(2a)
x = -(-2)/(2*1)
x = (2)/(2)
x = 1
To find the 'y', plug above value back into:
f(x) = x^2 – 2x + 1
f(1) = 1^2 – 2(1) + 1
f(1) = 1 – 2 + 1
f(1) = 0
Vertex: (1,0)
.
x-intercepts: set f(x) to zero and solve for x:
f(x) = x^2 – 2x + 1
0 = x^2 – 2x + 1
0 = (x-1)(x-1)
x = 1
x-intercept at (1,0) (same as the vertex, in this case)

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How do I write the equation of the line that has a y-intercept of (0,-6) and is perpendicular to the line 3x - 6y = 8.
.
Remember the "slope-intercept" form:
y = mx + b (equation of a line)
.
From:"the line that has a y-intercept of (0,-6)" we now know:
b = -6
.
To find 'm', begin by finding slope of:
3x - 6y = 8
-6y = -3x+8
6y = 3x-8
y = (3/6)x-(8/6)
y = (1/2)x-(4/3)
a line that is perpendicular to the above line MUST have a slope that is the negative reciprocal:
(1/2)m = -1
m = -2
.
Plug 'm' and 'b' back into:
y = mx + b
to get your answer:
y = -2x + (-6)
y = -2x - 6

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(x+4)(x-5)-(x+3)(x-6)
Yes, you would begin by FOILing:
(x^2-5x+4x-20)-(x^2-6x+3x-18)
(x^2-x-20)-(x^2-3x-18)
Now, distribute the "negative sign" to each term inside right parenthesis:
x^2-x-20-x^2+3x+18
-x-20+3x+18
2x-20+18
2x-2

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a ball is thrown from a 180m high cliff> the relation h=-5t^2-5t+180 relates the time in seconds to the height in m.
a) What was the intial height of the ball?
this is when t=0:
h=-5t^2-5t+180
h=-5(0)^2-5(0)+180
h=180 m
.
b) how long does it take the ball to reach a height of 80m?
set h to 80 and solve for t:
h=-5t^2-5t+180
80=-5t^2-5t+180
0=-5t^2-5t+100
0=-t^2-t+20
0=t^2+t-20
0=(t+5)(t-4)
t={-5,4}
throw out the negative solution leaving:
t = 4 seconds
.
c) determine the maximum height if the ball
this occurs at the vertex when
h=-5t^2-5t+180
t = -b/(2a)
t = -(-5)/(2(-5))
t = 5/(-10)
t = -1/2 seconds
.
Note: you probably typed in a wrong equation in your original problem
probably should be something like:
h=-5t^2+5t+180

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Cindy can paint a house in 3 hr less time than Dustin. Together, it takes 18 hrs. How long does it take each to do the job by theirselves?
.
Let x = time (hrs) it takes for Dustin
then
x-3 = time it takes for Cindy
.
18(1/x + 1/(x-3)) = 1
multiplying both sides by x(x-3):
18((x-3) + x) = x(x-3)
18(2x-3) = x^2-3x
36x-54 = x^2-3x
-54 = x^2-39x
0 = x^2-39x+54
solve by applying the "quadratic formula" to get:
x = {37.56, 1.44}
throw out the 1.44 (extraneous) to get
x = 37.56 hours (Dustin)
.
Cindy
x-3 = 37.56-3 = 34.56 hours
.
Details of quadratic follows:

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=1305 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 37.5623918681884, 1.43760813181156. Here's your graph:

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James is building a ramp for his elderly grandmother. If he builds it 3 feet above the ground and if it has a 4 foot base, how long will the ramp be?
A. 3 feet
B. 5 feet
C. 8 feet
D. 10 feet
.
It's a 3-4-5 triangle -- so it would be B.
Or, you can apply Pythagorean theorem:
Let r = ramp
then
r^2 = 3^2 + 4^2
r^2 = 9 + 16
r^2 = 25
r = 5 feet