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Section 5.3 Using Formulas

Subsection Volume and Surface Area

In Section 5.1 we used exponents to calculate the area of a square and the volume of a cube.

  • Volume is the amount of space contained within a three-dimensional object. It is measured in cubic units, such as cubic feet or cubic centimeters.
  • Surface area is the sum of the areas of all the faces or surfaces that contain a solid. It is measured in square units.

We saw that the area of a square whose side has length \(s\) units is given by \(A=s^2\) square units, and the volume of a cube of side \(s\) units is gven by \(V=s^3\) cubic units. Many other useful formulas involve exponents. The figure below shows several solid objects, along with formulas for their volumes and surface areas.

solids
Example 5.20.

At the Red Deer Pub and Microbrewery there is a spherical copper tank in which beer is brewed. If the tank is 4 feet in diameter, how much beer does it hold?

Solution

The formula for the volume of a sphere is \(V=\dfrac{4}{3} \pi r^3\text{,}\) where \(r\) is the radius of the sphere. The tank has diameter 4 feet, so its radius is 2 feet. We substitute \(r=\alert{2}\) into the formula, and simplify, following the order of operations. (If your calculator does not have a key for \(\pi\text{,}\) you can use the approximation \(\pi \approx 3.14\text{.}\))

\begin{align*} V \amp =\dfrac{4}{3} \pi (\alert{2})^3 \amp \amp \blert{\text{Compute the power first.}}\\ \amp = \dfrac{4 \pi (8)}{3} \amp \amp \blert{\text{Multiply by}~4\pi,~\text{divide by 3.}}\\ \amp=33.51 \ldots \end{align*}

The tank holds approximately 33.5 cubic feet of beer, or about 251 gallons.

Reading Questions Reading Questions

1.

What are the units of volume?

2.

What are the units of surface area?

Look Closer.

What does it mean for the volume of a round tank to be measured in "cubic" units? If we pour all the beer in the tank into a rectangular box, that box will hold 33 and a half cubes that measure 1 foot on each side.

Subsection Solving Equations with \(x^2\)

Taking a square root is the opposite of squaring a number. Thus, we can undo the squaring operation by taking square roots. For example, to solve the equation

\begin{equation*} x^2=64 \end{equation*}

we take the square root of each side. Saying that \(x^2\) equals 64 is the same as saying that \(x\) is a square root of 64. Remember that every positive number has two square roots, so we write

\begin{align*} x^2 \amp =64 \amp \amp \blert{\text{Take square roots of both sides.}}\\ \amp = \pm \sqrt{64} \\ \amp= \pm 8 \end{align*}

The equation has two solutions, \(8\) and \(-8\text{.}\)

Example 5.21.

The volume of a can of soup is 582 cubic centimeters, and its height is 10.5 centimeters. What is its radius?

Solution

The volume of a cylinder is given by the formula \(V=\pi r^2h\text{.}\) We substitute \(\alert{582}\) for \(V\) and \(\alert{10.5}\) for \(h\text{,}\) then solve for \(r\text{.}\)

\begin{align*} \alert{582} \amp = \pi r^2(\alert{10.5}) \amp \amp \blert{\text{Divide both sides by}~\pi.}\\ 185.256 \amp = r^2(10.5) \amp \amp \blert{\text{Divide both sides by 10.5.}}\\ 17.643 \amp = r^2 \amp \amp \blert{\text{Take square roots.}}\\ \pm 4.2 \amp = r \end{align*}

Because the radius of a soup can cannot be a negative number, we discard the negative solution in this application. The radius of the can is 4.2 centimeters.

The equations above, which involve the square of the variable (such as \(x^2\) or \(r^2\)), are called quadratic equations, and we shall see more about them later.

Reading Questions Reading Questions

3.

What operation is the opposite of squaring a number?

Subsection Pythagorean Theorem

A triangle in which one of the angles is a right angle, or \(90 \degree\text{,}\) is called a right triangle. The side opposite the right angle is the longest side of the triangle, and is called the hypotenuse. The other two sides of the triangle are called the legs.

If we know the lengths of any two sides of a right triangle, we can find the third side using a formula called the Pythagorean Theorem. We use the variable \(c\) for the length of the hypotenuse, and the lengths of the legs are denoted by \(a\) and \(b\) (it doesn't matter which is which).

Pythagorean Theorem.

If \(c\) stands for the length of the hypotenuse of a right triangle, and the lengths of the two legs are represented by \(a\) and \(b\text{,}\) then

\begin{equation*} a^2 + b^2 = c^2 \end{equation*}
right triangle

The Pythagorean theorem says that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs.

Example 5.22.

The length of a rectangle is 17 centimeters and its diagonal is 20 centimeters long. What is the width of the rectangle?

Solution

The diagonal of the rectangle is the hypotenuse of a right triangle, as shown in the figure. We are looking for the width of the rectangle, which forms one of the legs of the right triangle. We use the Pythagorean Theorem.

rectangle
\begin{align*} a^2 + b^2 \amp = c^2 \amp \amp \blert{\text{Substitute 20 for} ~c~ \text{and 17 for} ~b.}\\ a^2 + (\alert{17})^2 \amp = (\alert{20})^2 \end{align*}

We compute the powers, then solve for \(a\text{.}\)

\begin{align*} a^2 + 289 \amp = 400 \amp \amp \blert{\text{Subtract 289 from both sides.}}\\ a^2 \amp = 111 \amp \amp \blert{\text{Take square roots.}}\\ a \amp = \pm \sqrt{111} \end{align*}

We use a calculator to evaluate the square root and find that the width of the rectangle is approximately 10.536 centimeters.

Reading Questions Reading Questions

4.

What is a right triangle, and what is the hypotenuse?

5.

What do we use the Pythagorean theorem for?

Subsection Solving for One Variable

In the formula in Example 5.23, the variable \(P\) occurs more than once. We substitute the value for \(P\) each time it occurs.

Example 5.23.

If a principal of \(P\) dollars is invested at interest rate \(r\text{,}\) the formula

\begin{equation*} A = P + Prt \end{equation*}

gives the amount of money in an account after \(t\) years. (The "amount," \(A\text{,}\) is the original principal plus the interest earned.) How long will it take an investment of $5000 to grow to $7000 in an account that pays 6% interest?

Solution

We substitute \(\alert{7000}\) for \(A\text{,}\) \(\alert{5000}\) for \(P\text{,}\) and \(\alert{0.065}\) for \(r\) in the formula, and then solve the resulting equation for \(t\text{.}\)

\begin{align*} \alert{7000} \amp = \alert{5000} + \alert{5000}(\alert{0.065})t \amp \amp \blert{\text{Simplify the right side.}}\\ 7000 \amp = 5000+325t \amp \amp \blert{\text{Subtract 5000 from both sides.}}\\ 2000 \amp = 325t \amp \amp \blert{\text{Divide both sides by 325.}}\\ 615 \amp \approx t \end{align*}

However, the answer to the question in the problem is not 6.15 years! Because interest is paid only once a year, the investor must wait until the end of the seventh year before the account contains at least $7000.

Caution 5.24.

To simplify the right side of the equation

\begin{equation*} 7000 = 5000 + 5000(0.065)t \end{equation*}

it is not correct to add the two 5000's and write \(7000 = 10,000(0.065)t\text{.}\) Remember that multiplication comes before addition in the order of operations.

If we plan to use a formula more than once with different values for the variables, it may be faster in the long run to solve for one of the variables in terms of the others. We have already done something like this when we put a linear equation into slope-intercept form: in that situation we solved for \(y\) in terms of \(x\text{.}\)

Example 5.25.

Solve the formula \(~~A=P+Prt~~\) for \(t\text{.}\)

Solution

For this problem, we treat \(t\) as the unknown and treat all the other variables as if they were constants. We begin by isolating the term containing \(t\) on one side of the equation.

\begin{align*} A \amp = P+Prt \amp \amp \blert{\text{Subtract} ~P~ \text{from both sides.}}\\ A-P \amp = Prt \amp \amp \blert{\text{Divide bothe sides by}~Pr.}\\ \dfrac{A-P}{Pr} \amp = \dfrac{Prt}{Pr}\\ \dfrac{A-P}{Pr} \amp = t \end{align*}

We now have a new formula for \(t\) in terms of the other variables.

Caution 5.26.

In the expression \(P+Prt\) in Example 5.25, the terms \(P\) and \(Prt\) are not like terms. They cannot be combined.

Subsection Skills Warm-Up

Exercises Exercises

Follow the order of operations to simplify.

1.
\(3(-5)^2-2^3\)
2.
\(6(7-4)^2\)
3.
\(-6-2 \cdot 4^2\)
4.
\((5-3)^4(3-6)^3\)
5.
\(4(4-4^2)\)
6.
\(-3(-2)^2-5\)
7.
\(-(3 \cdot 2)^2-5\)
8.
\(3-4(-2)^3(-3)\)
9.
\(\dfrac{-5^2+1}{4}+\dfrac{2(-3)^3}{-6}\)
10.
\(\dfrac{2^2(-3^2)}{1-3^2}-\dfrac{7^2-6^2}{(1-3)^2}\)
11.
\(\dfrac{3^3-3}{(3-5)^3}-\dfrac{2(-2)^2-8}{-2^2(8-2^2)}\)
12.
\(3 \cdot \dfrac{5^3-(-10)^2}{3^2-3(5)} \cdot \dfrac{2^5+4}{3^2-2^2}\)

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 5.3

For Problems 1–2, write an expression for the shaded area in the combination of squares and circles.

1.
square
2.
circle

For Problems 3–4, write an expression for the volume of the figure.

3.
sphere
4.
box

For Problems 5–6, Find an expression for the surface area of each box. (Hint: Each box has six faces: top and bottom, front and back, left and right. Find the areas of all six faces, then add them.)

5.
box
6.
box

For Problems 7–11, find the unknown side or sides of the right triangles.

7.
triangle
8.
triangle
9.
triangle
10.
triangle
11.
triangle

For Problems 12–14, decide whether a triangle with the given sides is a right triangle.

12.

9 in, 16 in 25 in

13.

5 m, 12 m, 13 m

14.

\(5^2\) ft, \(8^2\) ft, \(13^2\) ft,

For Problems 15–18, explain why the equation is an incorrect application of the Pythagorean theorem for the figure.

15.
triangle

\(3^2+x^2 = 5^2\)

16.
right triangle

\(8+4=x\)

17.
right triangle

\(x^2+2^2=11^2\)

18.
right triangle

\(x^2+16=25\)

For Problems 19–22, solve the formula for the variable indicated.

19.

\(v=lwh~~~~~~~~\) for \(~w~\)

20.

\(E=\dfrac{mv^2}{2}~~~~~~~~\) for \(~m~\)

21.

\(A=\dfrac{h}{2}(b+c)~~~~~~~~\) for \(~h~\)

22.

\(F=\dfrac{9}{5}C+32~~~~~~~~\) for \(~C~\)

23.

\(A=\pi rh + 2\pi r^2~~~~~~~~\) for \(~h~\)

24.

\(\dfrac{x}{a}+\dfrac{y}{b}=1~~~~~~~~\) for \(~x~\)

For Problems 25–30,

  1. Assign a variable and and write an equation.
  2. Solve your equation and answer the question.
25.

Juliet's balcony is 24 feet above the ground, and there is a 10-foot moat at the base of the wall. How long a ladder will Romeo need to reach the balcony?

ladder
26.

Marlene visited the Quetzalcoatl pyramid near Mexico City last summer. She measured the base and found it is about 1400 feet on each side. She unrolled a ball of string as she climbed the face of the pyramid, and it was about 722 feet to the top. How tall is the pyramid?

pyramid
27.

A baseball diamond is a square whose sides are 90 feet long. What is the straight-line distance from home plate to second base?

28.

The light house is 5 miles east of Gravelly Point. The marina is 7 miles south of Gravelly Point. How far is it from the light house to the marina?

29.

A surveyor would like to know the distance across a lake. She picks a spot \(P\) on a line perpendicular to the width of the lake, and measures the two distances shown in the figure at right. How wide is the lake?

lake
30.

Find the length of each hypotenuse in the spiral figure shown at right.

spiral

Problems 31–33 involve an area, a circumference, or a volume. Decide which measure is appropriate, and then solve the problem.

31.

A circular pizza pan has a diameter of 14 inches. How much pizza dough is needed to cover it?

32.

To find the diameter of a large tree in his yard, Delbert wraps a string around the trunk and then measures its length. If the string is 72.25 inches long, what is the diameter of the tree?

33.

The first solo transatlantic balloon crossing was completed in 1984 in a helium-filled balloon called Rosie O'Grady. The diameter of the balloon was 58.7 feet. Assuming that the balloon was approximately spherical, calculate its volume.

34.

To calculate the area of a large circular fish pond, Graham measures the distance around the edge of the pond as 75.4 feet. Calculate the area of the pond.

Problems 35–36 use the formulas for volume and surface area.

35.

The dome on the new planetarium is a hemisphere (half a sphere) of radius 40 feet.

  1. Draw a sketch of the dome, and label its radius.
  2. How much space is enclosed within the dome?
  3. What is the surface area of the dome?
36.
  1. How much sheet steel is needed to make a cylindrical can with radius 5 centimeters and height 16 centimeters? (Hint: Should you calculate the volume or the surface area of the can?)
  2. How long is a cylindrical section of concrete pipe if its diameter is 3 feet and its surface area is approximately 848.25 square feet?
  3. Draw a sketch of the pipe in part (b), and label its dimensions.