2.3 Solve a Formula for a Specific Variable

Learning objectives

Learning objectives

• Solve a formula for a specific variable

• Use formulas to solve geometry applications

Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.

It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.

Geometric formulas often need to be solved for another variable, too. The formula $V=\frac{1}{3}\pi{r}^{2}h$ is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.

Example 2.28

Solution

We could now use this formula to find the height of a right circular cone when we know the volume and the radius of the base, by using the formula $h=\frac{3V}{\pi{r}^{2}}.$

In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.

Example 2.29

Solution

We can now use the formula $F=\frac{9}{5}C+32$ to find the Fahrenheit temperature when we know the Celsius temperature.

The next example uses the formula for the surface area of a right cylinder.

Example 2.30

Solution

Sometimes we might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.

Example 2.31

Solution

Use Formulas to Solve Geometry Applications

In this objective we will use some common geometry formulas. We will adapt our problem solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

When we solve geometry applications, we often have to use some of the properties of the figures. We will review those properties as needed.

The next example involves the area of a triangle. The area of a triangle is one-half the base times the height. We can write this as $A=\frac{1}{2}bh,$ where b = length of the base and h = height.

Example 2.32

Solution

In the next example, we will work with a right triangle. To solve for the measure of each angle, we need to use two triangle properties. In any triangle, the sum of the measures of the angles is $180^{\circ}.$ We can write this as a formula: $m\mathrm{∠}A+m\mathrm{∠}B+m\mathrm{∠}C=180.$ Also, since the triangle is a right triangle, we remember that a right triangle has one $90^{\circ}$ angle.

Here, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

Example 2.33

Solution

The next example uses another important geometry formula. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Writing the formula in every exercise and saying it aloud as you write it may help you memorize the Pythagorean Theorem.

We will use the Pythagorean Theorem in the next example.

Example 2.34

Solution

The next example is about the perimeter of a rectangle. Since the perimeter is just the distance around the rectangle, we find the sum of the lengths of its four sides—the sum of two lengths and two widths. We can write is as $P=2L+2W$ where L is the length and $W$ is the width. To solve the example, we will need to define the length in terms of the width.

Example 2.35

Solution

The next example is about the perimeter of a triangle. Since the perimeter is just the distance around the triangle, we find the sum of the lengths of its three sides. We can write this as $P=a+b+c,$ where a, b, and c are the lengths of the sides.

Example 2.36

Solution

Example 2.37

Solution

Applications of these geometric properties can be found in many everyday situations as shown in the next example.

Example 2.38

Solution

Section 2.3 Exercises

Section 2.3 Exercises

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

In the following exercises, solve for the formula for y.

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.

In the following exercises, solve using a geometry formula.

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Use the formula $A=\frac{1}{2}bh$ to solve for b.

2. Use the formula $A=\frac{1}{2}bh$ to solve for h.

3. Solve the formula $F=\frac{9}{5}C+32$ for C.

4. Solve the formula $A=\frac{1}{2}h(b+B)$ for b.

1. Solve the formula $A=P+Prt$ for t.

2. Solve the formula $A=P+Prt$ for r.

3. Solve the formula $4x+7y=9$ for y.

4. Solve the formula $5x+8y=1$ for y.

1. The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

2. A triangular tent door has an area of 15 square feet. The height is five feet. What is the length of the base?

3. The measure of one angle of a right triangle is 50 more than the measure of the smallest angle. Find the measures of all three angles.

4. The measure of one angle of a right triangle is 30 more than the measure of the smallest angle. Find the measures of all three angles.

1. Use the Pythagorean Theorem to find the length of the leg in the figure.

2. Use the Pythagorean Theorem to find the length of the leg in the figure.

3. The length of a rectangle is seven more than twice the width. The perimeter is 110 inches. Find the length and width.

4. The width of a rectangle is eight yards less than twice the length. The perimeter is 86 yards. Find the length and width.

1. One side of a triangle is seven inches more than the first side. The third side is four inches less than three times the first. The perimeter is 28 inches. Find the length of the three sides of the triangle.

2. One side of a triangle is three feet less than the first side. The third side is five feet less than twice the first. The perimeter is 20 feet. Find the length of the three sides of the triangle.

3. The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

4. The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

1. John puts the base of a 13-foot ladder five feet from the wall of his house as shown in the figure. How far up the wall does the ladder reach?

2. Randy wants to attach a 17-foot string of lights to the top of the 15 foot mast of his sailboat, as shown in the figure. How far from the base of the mast should he attach the end of the light string?

3. Solve the formula $C=\pi{}d$ for d.

4. Solve the formula $V=LWH$ for L.

1. Solve the formula $A=\frac{1}{2}bh$ for b.

2. Solve the formula$A=\frac{1}{2}{d}_{1}{d}_{2}$ for ${d}_{1}.$

3. Solve the formula$A=\frac{1}{2}h({b}_{1}+{b}_{2})$ for ${b}_{1}.$

4. Solve the formula$h=54t+\frac{1}{2}a{t}^{2}$ for a.

1. Solve $180=a+b+c$ for a.

2. Solve the formula$A=\frac{1}{2}pl+B$ for p.

3. Solve the formula$P=2L+2W$ for L.

4. Solve the formula$8x+y=15$ for y.

1. Solve the formula$-4x+y=-6$ for y.

2. Solve the formula$x-y=-4$ for y.

3. Solve the formula$4x+3y=7$ for y.

4. Solve the formula$2x+3y=12$ for y.

1. Solve the formula$3x-2y=18$ for y.

2. A triangular flag has area 0.75 square feet and height 1.5 foot. What is its base?

3. What is the base of a triangle with area 207 square inches and height 18 inches?

4. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

1. The angles in a triangle are such that one angle is twice the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

2. The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

3. The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

4. The perimeter of a rectangle of 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

1. The length of the rectangle is three meters less than twice the width. The perimeter of a rectangle is 36 meters. Find the dimensions of the rectangle.

2. The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet longer than the second side. Find the length of each side.

3. One side of a triangle is twice the smallest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

4. The perimeter of a rectangular field is 560 yards. The length is 40 yards more than the width. Find the length and width of the field.

1. A rectangular parking lot has perimeter 250 feet. The length is five feet more than twice the width. Find the length and width of the parking lot.

2. A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display as shown. How far from the base of the pole should the end of the string of lights be anchored?

3. Chi is planning to put a diagonal path of paving stones through her flower garden as shown. The flower garden is a square with side 10 feet. What will the length of the path be?

4. Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40° Celsius. Solve for F in the formula $C=\frac{5}{9}(F-32)$ to find the Fahrenheit temperature.

1. Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

2. If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

3. Look at the two figures below.

ⓐ Which figure looks like it has the larger area? Which looks like it has the larger perimeter?ⓑ Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?ⓒ Were the results of part (b) the same as your answers in part (a)? Is that surprising to you?