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see the attached worksheet and the comments below it.



first you change the division into a multiplication because:
(a/b) / (c/d) = (a/b) * (d/c)
then you multiply the first fraction by 2n/2n and the second fraction by n/n.
you are using the associative law of multiplication to do this.
after this is done yuo are left with fractions that can be factored further to simplify them.
once simplified, the common terms in the numerator and the denominator can be canceled out and you are left with the simplified answer.
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see the attached worksheet and the words underneath.

first you change this into a multiplication problem because:
(a/b) / (c/d) = (a/b) * (d/c)
then you multiply the first fraction by a/a and you multiply the second fraction by b/b.
this removes the fraction parts of the numerators and denominators or each.
you then remove the parts of the numerator that cancel out with equal parts in the denominator.
the result is your answer.
you are using the associative law of multiplication when you multiply the first fraction by a/a and the second fraction b/b.
the associative law of multiplication says you can do the following:
a * b * c = (a * b) * c = a * (b * c)
a * b * c * d = (a * b) * (c * d)

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see the attached worksheet

if you multiply by x^2 / x^2, you will remove the denominators and you will be left with a quadratic equation divided by another quadratic equation which, when factored, will allow some of the factors to cancel out and you are then left with the simplified solution which is your answer.

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you want to divide 4x^2 - 15 by (2x - 5)
you need to fill in the missing orders of exponents.
when you do that, your equation you are dividing into will be equal to:
4x^2 + 0x - 15
you want to divide that by (2x - 5)
you start by dividing 2x into 4x^2 to get 2x because 2x * 2x = 4x^2
you then multiply (2x - 5) by 2x to get 4x^2 - 10x
you then subtract 4x^2 - 10x from 4x^2 + 0x - 15 to get 10x - 15
you then divide 2x into 10x to get 5 because 5 * 2x = 10x
you then multiply (2x - 5) by 5 to get 10x - 25
you then subtract 10x - 25 from 10x - 15 to get + 10
since + 10 is of a lower order than 2x, you are done and your remainder is 10.
your quotient is 2x + 5 + 10/(2x - 5)

instructions on how to do polynomial division can be found at the following website.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut36_longdiv.htm

instructions on how to do synthetic division can be found at the following website.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut37_syndiv.htm

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you are dividing 4x^2 - 3x + 2 by x
divide 4x^2 by x to get 4x because x * 4x = 4x^2
subtract 4x^2 from 4x^2 - 3x + 2 to get -3x + 2
divide -3x by x to get -3 because -3 * x = -3x
subtract -3x from -3x + 2 to get + 2
since x is a higher order than 2 you are done dividing and 2 is your remainder.
your quotient is 4x - 3 + (2/x)
if you multiply that by x you will get back to 4x^2 - 3x + 2.

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this factors to:

2x * (4x - 3) divided by 2x * 2x

the 2x in the numerator and denominator cancel out and you are left with:

4x - 3 divided by 2x

that's the simplified expression.

the restrictions come from the original factored equation which is:

2x * (4x - 3) divided by 2x * 2x

x cannot be equal to 0, otherwise the denominator becomes equal to 0.

otherwise you're ok.

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this factors to:

x * (x^2 + 6x + 9) divided by x * (x^2 - 9) which factors to:

x * (x + 3) * (x + 3) divided by x * (x + 3) * (x - 3)

The x in the numerator and denominator cancel out.

The (x + 3) in the numerator and denominator cancel out.

you are left with:

(x + 3) / (x - 3)

that's the simplified equation.

the restrictions come from the original factored equation, which is:

x * (x + 3) * (x + 3) divided by x * (x + 3) * (x - 3)

x in the denominator can't be equal to 0.

x can't be equal to -3, otherwise (x + 3) in the denominator would be equal to 0.
x can't be equal to 3, otherwise (x - 3) in the denominator would be equal to 0.

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since (x+3)^-1 is equivalent to 1 / (x + 3)^2, this equation translates to:

(x^2 + x - 6) divided by (x + 3)^2

this factors to:

(x + 3) * (x - 2) divided by (x + 3) * (x + 3)

(x + 3) in the numerator and denominator cancel out and you are left with:

(x - 2) / (x + 3)

that's the simplified equation.

the restrictions come from the original equation after it has been factored.

that equation is:

(x + 3) * (x - 2) divided by (x + 3) * (x + 3)

x cannot be equal to -3 because (x + 3) = 0.

otherwise you're ok.

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this factors to:

(x-y)(x+y) divided by (x+2y)(x+y)

the (x+y) in the numerator and denominator cancel out and you are left with:

(x-y) divided by (x + 2y)

that's the simplified expression.

the restrictions come from the original equation.

the original equation is:

(x^2 - y^2) divided by (x^2 + 3xy + 2y^2) which factored out to be:

(x - y) * (x + y) divided by (x + y) * (x + 2y)

since the denominator can't be equal to 0, then the following restrictions apply.

x cannot be equal to -y, otherwise (x + y) will be equal to 0.

x cannot be equal to -2y, otherwise (x + 2y) will be equal to 0.

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it appears that if you add 100 ml of a 0% solution, then you will have 200 ml of a 3% solution.

6% solution in 100 ml is equal to 6 ml because 6 ml divided by 100 ml = .06 which is equivalent to 6%.

6 ml divided by 200 ml is equal to .03 which is equivalent to 3%.

the formula that can be used is:

x = the amount of 0% solution you need to add.

you get .06 * 100 + 0 * x = .03 * (100 + x)

you are adding 100 ml of 6% solution to x ml of 0% solution to get a total of (100 + x) ml of 3% solution.

the formula becomes:

.06 * 100 + 0 * x = .03 * (100 + x)

simplify to get:

.06 * 100 + 0 * x = .03 * 100 + .03 * x

simplify further to get:

6 = 3 + .03 * x

subtract 3 from both sides of the equation to get:

3 = .03 * x

divide both sides of the equation by .03 to get:

3 /.03 = x

simplify to get:

100 = x.

you need to add 100 ml of 0% solution to 100 ml of 6% solution to get 200 ml of 3% solution.

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you have a cone and a half of a sphere sitting on top of the cone.

the base of the cone is a circle with a radius of 12.5

the base of the hemisphere is a circle that is the same size as the base of the cone.

The equations for volume and surface area of a sphere and a cone can be found in the following link:

http://math.about.com/od/formulas/ss/surfaceareavol.htm

for the sphere:
volume = 4/3 * pi * r^3
surface area = 4 * pi * r^2
r = radius

for the cone:
volume = 1/3 * pi * r^2 * h
area of base = pi * r^2
lateral surface area = pi * r * s
h = height
r = radius
s = slant height

you are given that the surface area of the hemisphere is the same as the surface area of the cone.

the surface area of the hemisphere is 1/2 the surface area of the sphere.
the formula for the surface area of the hemisphere is therefore equal to:

1/2 * 4 * pi * r^2 which is equal to 2 * pi * r^2.

The lateral surface area of the cone is equal to pi * r * s

since the surface area of the hemisphere is equal to the lateral surface area of the cone, you get:

2 * pi * r^2 = pi * r * s

since r is equal to 12.5, this formula becomes:

2 * pi * (12.5)^2 = pi * 12.5 * s

you can solve for s to get:

s = (2 * pi * (12.5)^2) / (pi * 12.5)

after cancelling of common terms in the numerator and denominator, this formula becomes:

s = 2 * 12.5 which is equal to 25.

the slant height of the cone is equal to 25.

now that you found the slant height of the cone, you can find the height of the cone.

the slant height of the cone and the height of the cone and the radius of the cone form a right triangle with one leg equal to 12.5 and the hypotenuse equal to 25.

the pythagorean formula of a^2 + b^2 = c^2 can be used to find the height.

let a = 12.5 and c = 25 and b = the height of the cone.

you get (12.5)^2 + b^2 = 25^2 which becomes:

156.25 + b^2 = 625

solve for b to get b = sqrt (625 - 156.25) = 21.65063509.

that's the height of the cone.

now that you know the height of the cone, you can find the volume of the cone.

the volume of the cone is equal to 1/3 * pi * r^2 * h.

this is equal to 1/3 * pi * (12.5)^2 * 21.65063509 which becomes:

volume of the cone = 3542.576883 cubic units.

volume of the hemisphere is equal to 1/2 the volume of the sphere which is equal to 1/2 * 4/3 * pi * r^3 which becomes:

volume of the hemisphere = 4090.615434 cubic units.

add the 2 volumes together and you get a total volume of 7633.192310 cubic units.

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assuming the number of days in the year is 365, the calculations are done as follows:

annual percentage rate = 2.5%
nominal annual interest rate = apr / 100% = .025
number of compounding periods per year = 365
daily interest rate = .025 / 365 = .00006849315068
add 1 to that to get 1.00006849315068
raise that to the 365th power to get (1.00006849315068)^365 = 1.025314243
subtract 1 from that to get .025314243
that's the effective annual interest rate.
multiply that by 100% to get 2.5314243%
that's the annual percentage yield.

if you invest a dollar today and you get 2.5% annual percentage rate, you will have 1.025314243 dollars at the end of the year.

your effective interest rate is equal to .025314243
your annual percentage yield is 2.5314243%.

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read the following:

There are 3 basic methods, not including the indian method, which can be considered a 4th method not widely used.
1. factoring
2. sum of squares
3. quadratic formula

the one formula that works all the time is the quadratic formula.

this formula can be found here:
http://www.purplemath.com/modules/quadform.htm

to use this formula, you have to convert the quadratic equation into standard form.

standard form is:

ax^2 + bx + c = 0

a is the coefficient of the x^2 term.
b is the coefficient of the x term.
c is the constant term.

once you do that you can find the value of a, b, and c and then plug it into the quadratic formula to get your answer.

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slope intercept form is y = mx + b
m is the slope
b is the y intercept

standard form is ax + by = c
a is the coefficient of x
b is the coefficient of y
c is the constant.

to convert from slope intercept form to standard form, you need to get both the x and the y on the left side of the equation.

assuming your equation is y = -x - 12, you would do the following:

add x to both sides of the equation to get:
x + y = -12

in this equation, that's all you have to do.
you are in the form of:
ax + by = c
a is equal to 1
b is equal to 1
c is equal to -12

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Let x = the probability of a non-smoker to die of lung cancer.
Let 15 * x = the probability of a smoker to die of lung cancer.
Let w * x = the probability of a smoker who quit t years ago to die of lung cancer.
you are given that w = 15 - t.

Polly never smoked.
Sandy quit 2.5 years ago.

Since w = 15 - t, and t = 2.5, this means that the probability that Sandy will die of lung cancer is 12.5 * the probability that Polly will die of lung cancer.

To graph the equation, let x = the number of years that the smoker has quit smoking.

let y = w which is equal to 15 - x.

The graph will look like this:



From the graph, you can see that when x = 2.5, y = 12.5.

It's actually hard to see, but you can do the calculations to prove it as follows:

w = 15 - t and t = 2.5 which makes w = 15 - 2.5 which is equal to 12.5.

on the graph, y is equal to w and x is equal to t.

if you can graph a horizontal line at y = 12.5 (shown on the graph), and a vertical line at x = 2.5 (approximated on the graph), then the intersection of the vertical and horizontal line should be intersecting the graph of y = 15 - x at about the same point, which it does.

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the mean is 30 inches.

the standard deviation is 3/4 inch.

to find p(x > 32) you need to calculate the z-score to get:

z = (32 - 30) / (3/4) which is equal to 8/3.

that's a z-score of 2.67 rounding to two decimal places.

the formula to find the z-score is:

z-score = (value - mean) / (standard deviation)

look up in the z-score tables for a z-score of 2.67 and you will find that the probability of the z-score being less than 2.67 is equal to .9962.

The probability of the z-score being greater than 2.67 is equal to 1 - .9962 which is equal to .0038.

the z-score table that i used is shown below:

http://lilt.ilstu.edu/dasacke/eco148/ztable.htm

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The surface area of a cylinder is equal to the lateral area plus the area of the top and bottom base.

The area of each base of the cylinder is equal to pi * r^2 which is equal to pi * (4.7)^2 which is equal to pi * 22.09 which is equal to 22.09 * pi.

multiply this by 2 to get a total of 44.18 * pi

the lateral area of the cylinder is equal to 2 * pi * r * h which is equal to 2 * pi * 4.7 * 12 which is equal to 112.8 * pi

add up 44.18 * pi and 112.8 * pi and you get a total surface area of 156.98 * pi.

this is the surface area in terms of pi.

multiply by the value of pi to get:

total surface area equals 493.1672148 square meters which can be rounded to however many decimal places you require.

here's a reference you can look at to see what these formulas mean.

http://www.mathopenref.com/cylinderarea.html

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$0.42 * 100 = 42 cents.

if 3 melons cost 42 cents, then the cost per melon is equal to 42 / 3 = 14 cents.

7 melons times the cost per melon of 14 cents would equal 7 * 14 which would be equal to 98 cents.

98 cents / 100 = $0.98.

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if p = 19, then 745 + 5p = 745 + 5*19 = 745 + 95 = 840
you just replace p with 19 in the expression and then simplify the expression to get the answer.

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let x represent the number.

3 multiplied by the sum of 4 and the number is the same as 18 more than the number.

this statement translates to:

3 * (x + 4) = x + 18

solve for x as follows:

simplify the equation to get:
3x + 12 = x + 18
subtract 12 from both sides of the equation to get:
3x = x + 6
subtract x from both sides of the equation to get:
2x = 6
divide both sides of the equation by 2 to get:
x = 3

that should be your answer.

substitute the value of 3 for the number to see if this solves the original problem.

the original problem is:

Three multiplied by the sum of 4 and a number is the same as 18 more than the number. Find the number.

Assuming the number is 3, this statement then becomes:

Three multiplied by the sum of 4 and 3 is the same as 18 more than 3. Find the number.

18 more than 3 is equal to 21.

3 multiplied by the sum of 4 and 3 is 3 multiplied by 7 which is equal to 21.

This confirms the solution that the number is 3.

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The domain appears to be all values of x except x = -1 and x = 0
The range appears to be all values of y.
The zeroes (roots) appear to be x = -2/3 and x = 1.

you find the domain by looking for all values of x that will result in a value of y.

all values of x will result in a value of y except when x = -1 and when x = 0.

when x = -1 or when x = 0, the denominator of the original equation will be equal to 0 which makes the value of y undefined.

therefore x = -1 and x = 0 can't be in the domain.

since there are no further restrictions as to what the value of x can be, then the domain is all values of x except at x = -1 and x = 0.

the zeroes of the equation are found by setting the equation equal to 0 and solving for x.
it's best to simplify the equation first.

start with (3x^3 - x^2 - 2x) / (x * (x+1)^2

since (3x^3 - x^2 - 2x) / x is equal to (3x^2 - x - 2), the equation becomes:

(3x^2 - x - 2) / (x+1)^2

set this equal to 0 to get:

(3x^2 - x - 2) / (x + 1)^2 = 0

multiply both sides of the equation by (x + 1)^2 to get:

(3x^2 - x - 2) = 0

factor to get:

(3x + 2) (x - 1) = 0

solve for x to get:

x = -2/3 or x = 1

these are the roots of the quadratic equation which is the point at which the equation crosses the x-axis which is the zeroes of the equation.

you can confirm that the solutions are correct by graphing the equation.

the graph of the equation looks like this:



from this graph, you can see that the zeroes of the graph are around x = -2/3 and x = 1.

from this graph, you can see that there appears to be an asymptote at around x = -1 although it's not real easy to see that.

from this graph, you cannot see that there is a hole at x = 0

the value of y is undefined at x = -1 and also undefined at x = 0.

this is because the denominator of the equation is 0 at those values of x.

at x = -1, this results in an asymptote.

at x = 0, this results in a hole.

if you try to find the value of y when x = 0 and when x = -1, you will not be able to.

you can find a value of y for any other value of x other than 0 or -1.

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answers are shown below:

A rectangular box measures 18 inches wide, 15 inches long, and 10 inches high.

-----

A.) What is the volume of the box in cubic inches? In cubic feet?

since the measurements of the box are already in inches, then the volume is equal to 18 * 15 * 10 which is equal to 2700 cubic inches.

you need to convert inches to feet and then do the calculation again.

since 18 inches / 12 = 1.5 feet and 15 inches / 12 = 1.25 feet and 10 inches / 12 = .833333333 feet, your volume is equal to 1.5 * 1.25 * .833333333 which is equal to 1.5625 cubic feet.

alternatively, you could have taken the number of cubic inches and divided them by the number of cubic inches per cubic foot.

the number of cubic inches per cubic foot is equal to 12 * 12 * 12 which is equal to 1728 cubic inches per cubic foot.

your original answer in cubic inches was 2700 cubic inches.

divide 2700 cubic inches by 1728 cubic inches per cubic foot and you get 1.5625 cubic feet.

you get the same answer derived in a different, but equivalent, way.

-----

B.) A big bag of packing peanuts says that it holds 1.2 cubic feet of peanuts. How much more peanuts are needed to fill the box, or how much extra is there?

since the volume of the box is 1.5625 cubic feet and the volume of the peanuts is equal to 1.2 cubic feet, then the difference in cubic feet of peanuts is required to fill the box.

1.5625 - 1.2 cubic feet equals .3625 cubic feet.

this means you need another .3625 cubic feet of peanuts to fill the box.

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C.) What part(fraction) of a box will the peanuts fill, or how many boxes of this size will the peanuts fill? Express the answer as a fraction, then as a decimal.

the peanuts will fill 1.2 / 1.5625 = .768 of the volume of the box.

the decimal is .768

the fraction is 1.2 / 1.5625 which i was able to simplify to 96 / 125.

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D.) How high up in the box will the peanuts come?

to find this, you need to freeze the length and the width of the box and then solve for the height using the volume of the peanuts.

the length of the box is 1.2 feet and width of the box is 1.5 feet.

the volume of the peanuts is 1.2 cubic feet.

length * width * height = volume

this translates to:

1.25 * 1.5 * height = 1.2 cubic feet.

simplify this equation to get:

1.875 square feet * height = 1.2 cubic feet

divide both sides of this equation by 1.875 to get:

height = 1.2 cubic feet divided by 1.875 square feet = .64 feet.

the height of the peanuts in the box is equal to .64 feet.

to confirm, calculate the volume of the peanuts in the box.

the volume of the peanuts in the box is equal to 1.25 * 1.5 * .64 which is equal to 1.2 cubic feet.

you needed to freeze the length and the width and allow the height to vary.

-----

E.) Explain how questions b, c, and d are similar and how are they different.

questions b, c, and d all involve the volume of the peanuts in relationship to the volume of the box.

b asks for the volume of the peanuts in the box.
c asks for the ratio of the volume of the peanuts in the box to the volume of the box.
d asks for the height of the peanuts in the box which is a function of the formula for the volume of the peanuts in the box.

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it costs 2200 to manufacture 100 chairs in a day.
it costs 4800 to manufacture 300 chairs in one day.

since there is a linear relationship, your relationship will be on a straight line.

you want to find the slope of this straight line and you want to find the y-intercept of this straight line.

to graph this relationship, you will want to show the number of chairs manufactured on the x-axis and the cost on the y-axis.

the slope of your line is equal to the formula:

slope = (y2 - y1) / (x2 - x1)

since it costs 2200 to manufacture 100 chairs, assign the value of 100 chairs to x1 and the value of 2200 dollars to y1.

since it costs 4800 to manufacture 300 chairs, assign the value of 300 chairs to x2 and the value of 4800 dollars to y2.

now you have the following values that you can use to solve for the slope.

you have:

x1 = 100
y1 = 2200

x2 = 300
y2 = 4800

you slope formuls of (y2 - y1) / (x2 - x1) becomes:

slope = (4800 - 2200) / (300 - 100)

this simplifies to:

slope = 2600 / 200 which simplifies further to 13 / 1 which is equal to 13.

you now have the slope of the equation of your line and all you need is the y-intercept.

this is the value of y when x is equal to 0.

to find the y-intercept, you can use the point slope form of the equation for a straight line.

the point slope form is:

y = y1 = m * (x - x1)

(x1,y1) can be any point on the line.

we'll assign the value of (100,2200) to the point (x1,y1).

this means that:

x1 = 100
y1 = 2200

m is equal to the slope which we have already figured out is equal to 13.

the point slope form of the equation becomes:

y - 2200 = 13 * (x - 100)

we solve this equation for y to get:

y = 13 * (x - 100) + 2200.

we simplify this equation to get:

y = 13x - 1300 + 2200.

we simplify this equation further to getr:

y = 13x + 900.

the equation i now in the slope intercept form of the equation for a straight line.

that form is y = m*x + b where m is the slope and b is the y-intercept.

consequently, the slope is equal to 13 and the y-intercept is equal to 900.

the y-intercept is the value of y when the value of x is equal to 0.

we can graph the equation of y = 13x + 900 as shown below:



your equation is:

y = 13x + 900

you can see from the graph that, when x = 100, y = 2200, and when x = 300, y = 4800.

this agrees with the problem statement, so the graph is accurate which means that the equation is accurate.

when x = 0, you can also see from the graph that the value of y is equal to 900.

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i will use the letter T to represent theta.

the pythagorean identify is sin^2(T) + cos^2(T) = 1

subtract sin^2(T) from both sides of this equation to get:

cos^2(T) = 1 - sin^2(T)

your equation is 1 - sin^2(T) = .5

since 1 - sin^2(T) is equivalent to cos^2(T), replace 1 - sin^2(T) with cos^2(T) and you get:

cos^2(T) = .5

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

cos(T) = +/- sqrt(.5)

cos(T) can be + sqrt(.5) or - sqrt(.5)

if + sqrt(.5), then T is equal to 45 degrees.

if - sqrt(.5), then T is equal to 135 degrees.

to find the angle whose cosine is equal to sqrt(.5), you use the cos^-1 function of your calculator.

the key to solving this, however, is to recognize that:
1 - sin^2(T)
is equivalent to cos^2(T) based on the pythagorean formula of:
sin^2(T) + cost^2(T) = 1.

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let T equal the tens digit.
let U equal the units digit.

you are given that the sum of its digits is 3 times its units digit.

this translates to the following equation:

T + U = 3 * U

you are also given that its tens digit exceeds its units digit by 4.

this translates to the following equation:

T = U + 4

from these relationships, you can figure out what value each of the digits is.

since T = U + 4, you can replace T with U + 4 in the first equation of T + U = 3 * U.

T + U = 3 * U becomes:

(U + 4) + U = 3 * U

sinplify this equation to get:

U + 4 + U = 3 * U

combine like terms to get:

2 * U + 4 = 3 * U

subtract 2 * U from both side of this equation to get:

4 = U

the units digit is equal to 4.

from the equation T = U + 4, you can solve for T to get:

T = 4 + 4 = 8

you now have:

T = 8
U = 4

that's your solution.

it translates to:

the tens digit is equal to 8 and the units digit is equal to 4.

you can confirm by using those values in the original problem statement to see if the original problem statement is true.

the original problem statement is:

A two digit number the sum of its two digits is three times of its units digit and its tens digit exceeds its units digit by 4.

the 2 digit number is 84.

the sum of 8 and 4 is equal to 12 which is 3 times the unit digit of 4.

the tens digit is 8 which exceeds the unit digit of 4 by 4.

both problem statements are true when you assume 8 for the tens digit and 4 for the units digit so that's your answer.

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the attached reference give some insight into the scale on a map.
http://egsc.usgs.gov/isb/pubs/factsheets/fs01502.pdf

assuming they are talking about inches, then the scale can be interpreted as follows:

1 inch on the map represents 20,000 inches in the real world.

1 km in the real world is equivalent to 39,370.1 inches.

2 km in the real world would be equivalent to 2 * 39,370.1 which would be equal to 78,740.2 inches.

2 km in the real world is equivalent to 78740.2 inches.

the ratio to the map is:

1 / 20,000 = x / 78,740.2

solve for x and you get x = 78,740.2 / 20,000 = 3.93701 inches.

3.93701 inches on the map represents 2 kilometers in the real world.

assuming they are talking about centimeters, then the ratio of 1:20,000 would be analyzed as follows:

1 km in the real world is equivalent to 100,000 centimeters.

2 km in the real world is therefore equivalent to 200,000 centimeters.

with a scale 1 centimeter on the map to 20,000 centimeters in the real world, the equivalent measure on the map would be:

200,000 / 20,000 = 10 centimeters on the map.

we assumed inches and we got 3.93701 inches on the map is equivalent to 2 kilometers in the real world.

we assumed centimeters and we got 10 centimeters on the map is equivalent to 2 kilometers in the real world.

since an inch is equivalent to 2.54 centimeters, we can convert the measurement on the map from centimeters to inches by dividing by 2.54.

when we do that, 10 centimeters on the map is equivalent to 3.93701 inches on the map.

in other words, we got the same measure on the map whether or not we assumed inches or centimeters.

3.93701 inches is equivalent to 10 centimeters.

if we assumed inches, we would measure 3.93701 inches.
if we assumed centimeters, we would measure 10 centimeters.

the actual distance on the map would be the same.

to answer the second question you posed, centimeters on the map is assumed.

5 centimeters on the map is equivalent to how many kilometers in the real world.

the scale of the map is 1:5,000

this means that 1 cm on the map is equivalent to 5,000 cm in the real world.

if the distance on the map is 5 cm, then the distance in the real world is 5 * 5,000 cm which is equal to 25,000 cm.

since 1 km is equivalent to 100,000 cm in the real world, then 25,000 cm is equivalent to 25,000 / 100,000 = .25 kilometers.

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the formula you are using is area of a triangle is equal to 1/2 * base * height.
the base is a
the height is b * sin(c)
my diagram of your triangle is shown below:

in the diagram, h is the height and a is the base.
using trigonometry, we get:
sin(c) = oppositve / hypotenuse = h / b
multiplying both sides of this equation by b, we get:
h = b * sin(c)
since they're equal, we can substitute one for the other.
the formula is area = 1/2 * base * height.
the base is equal to a.
the height is equal to b * sin(c)
the formula becomes:
area = 1/2 * a * b * sin(c)
to find the area of the triangle using this formula, you need to know what angle C is.

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it appears that you have an isosceles triangle with an altitude dropping from vertex A to intersect with the line BC at D.
the isosceles triangle is ABC.
The altitude drops to intersect with line BC at D.
this forms 2 triangles.
they are triangle ABD and triangle ACD
you are given the line AB is congruent to line AC which is why the triangle ABC is isosceles.
you are given that angle BAD is congruent to angle CAD which means that the altitude AD is the bisector of angle BAC.
since triangle BAC is an isosceles triangle and side AB is congruent to side AC, this means that angle ABD is congruent to ACD.
this means that triangle ABD is congruent to triangle ABD by angle side angle (ASA).
this means that side AD is congruent to side BC because corresponding sides of congruent triangles are congruent.
my diagrm of your triangle is shown below:

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you can't find the base length unless you know the vertex angle.
that's the angle between the 2 equal sides.
the 2 equal sides are 8 feet long each.
the base will be any lengthy depending on the angle between the legs.
it will range from 0 to 16
it can't be equal to 0 because then your 2 legs will have collapsed against each other and there is no space left for the garden.
it can't be 180 degrees because the legs will have collapsed against the base and you will not have any space for the garden.
the angle has to be somewhere between 0 and 180 degrees but not equal to 0 or 180 degrees.
assuming an angle of 90 degrees, then your isosceles triangle will have 2 legs of 8 feet each and a base of with a length of square root of (128)
this is because by making the vertex angle 90 degrees, you created a right triangle with legs of 8 feet each and a hypotenuse as the base which means the pythagorean triangle formula can be used which is:
a^2 + b^2 = c^2
one of the legs if a and one of the legs is b and c is the base / hypotenuse of the isosceles triangle.
you get 8^2 + 8^2 = c^2 which results in 64 + 64 = c^2 which results in 128 = c^2 which results in c = square root of 128.
that's the length of the base, assuming the vertex angle is 90 degrees.
that length is equal to 11.3237085 in decimal format.
remember now, the vertex angle is the angle between the equal legs of the isosceles triangle.
change the angle, and the length of the base changes.

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let x = your age now.
your brother's age is x - 18 now because x - 18 = 10
your mother's age is x + 22 now because x + 22 = 50
you want to find out how old each of you will be when the sum of your ages is 150.
the equation would be x + x - 18 + x + 22 = 150
combine like terms to get 3x + 4 = 150
subtract 4 from both sides of the equation to get 3x = 146
divide both sides of the equation by 3 to get x = 146/3
when the sum of your ages is 150, you will be 146/3 years old, your brother will be 146/3 - 18 years old, and your mother will be 146/3 + 22 years old.
in decimal terms, that means:
you will be 48.67 years old
your brother will be 30.67 years old
your mother will be 70.67 years old
add them up and they equal 150.01 which is the same as 150 if we ignore the rounding error.
the older person is 50 years old.
you are 28.
when you were born, the older person was 28 years younger which makes the older person 50 - 28 = 22.
your mother was 22 when you were born.
your age is now x.
your mother's age is x + 22.
when x is equal to 0, x + 22 is equal to 22 so it check out.
the reasoning behind this is that we are taking your age as the base and everything else is relative to that base.
the difference between their age and your is constant which allows the equatins to work regardless of your age.

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i appears there's a vertical asymptote at x = -1.
the following table helps show that.

x	y = log(2,x+1)
-2	#NUM!
-1	#NUM!
-0.9999	-16.60964047
-0.9	-3.321928095
-0.8	-2.321928095
-0.7	-1.736965594
-0.6	-1.321928095
-0.5	-1
0	0
1	1
2	1.584962501
3	2
4	2.321928095
5	2.584962501
6	2.807354922
7	3
8	3.169925001
9	3.321928095

you cannot take a log of a negative number of or 0.
that is why the first 2 values in the table are invalid.
that is why the graph stops as just to the right of x -1
log (x+1) will be log(0) when x is equal to -1.