Practice Makes Perfect
Solve a Formula for a Specific Variable
In the following exercises, solve the given formula for the specified variable.
1. Solve the formula \(C=πd\) for \(d\).
 Answer

\(d=\dfrac{C}{π}\)
2. Solve the formula \(C=πd\) for \(π\).
3. Solve the formula \(V=LWH\) for \(L\).
 Answer

\(L=\dfrac{V}{WH}\)
4. Solve the formula \(V=LWH\) for \(H\).
5. Solve the formula \(A=\frac{1}{2}bh\) for \(b\).
 Answer

\(b=\dfrac{2A}{h}\)
6. Solve the formula \(A=\frac{1}{2}bh\) for \(h\).
7. Solve the formula
\(A=\frac{1}{2}d_1d_2\) for \(d_1\).
 Answer

\(d_1=\dfrac{2A}{d_2}\)
8. Solve the formula
\(A=\frac{1}{2}d_1d_2\) for \(d_2.\)
9. Solve the formula
\(A=\frac{1}{2}h(b_1+b_2)\) for \(b_1\).
 Answer

\(b_1=\dfrac{2A}{h}−b_2\)
10. Solve the formula
\(A=\frac{1}{2}h(b_1+b_2)\) for \(b_2\).
11. Solve the formula
\(h=54t+\frac{1}{2}at^2\) for \(a\).
 Answer

\(a=\dfrac{2h−108t}{t^2}\)
12. Solve the formula
\(h=48t+\frac{1}{2}at^2\) for \(a\).
13. Solve \(180=a+b+c\) for \(a\).
 Answer

\(a=180−b−c\)
14. Solve \(180=a+b+c\) for \(c\).
15. Solve the formula
\(A=\frac{1}{2}pI+B\) for \(p\).
 Answer

\(p=\dfrac{2A−2B}{I}\)
16. Solve the formula
\(A=\frac{1}{2}pI+B\) for \(I\).
17. Solve the formula
\(P=2L+2W\) for \(L\).
 Answer

\(L=\dfrac{P−2W}{2}\)
18. Solve the formula
\(P=2L+2W\) for \(W\).
In the following exercises, solve for the formula for \(y\).
19. Solve the formula
\(8x+y=15\) for \(y\).
 Answer

\(y=15−8x\)
20. Solve the formula
\(9x+y=13\) for \(y\).
21. Solve the formula
\(−4x+y=−6\) for \(y\).
 Answer

\(y=−6+4x\)
22. Solve the formula
\(−5x+y=−1\) for \(y\).
23. Solve the formula
\(x−y=−4\) for \(y\).
 Answer

\(y=4+x\)
24. Solve the formula
\(x−y=−3\) for \(y\).
25. Solve the formula
\(4x+3y=7\) for \(y\).
 Answer

\(y=\frac{7−4x}{3}\)
26. Solve the formula
\(3x+2y=11\) for \(y\).
27. Solve the formula
\(2x+3y=12\) for \(y\).
 Answer

\(y=\frac{12−2x}{3}\)
28. Solve the formula
\(5x+2y=10\) for \(y\).
29. Solve the formula
\(3x−2y=18\) for \(y\).
 Answer

\(y=\frac{18−3x}{−2}\)
30. Solve the formula
\(4x−3y=12\) for \(y\).
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.
31. A triangular flag has area 0.75 square feet and height 1.5 foot. What is its base?
 Answer

1 foot
32. A triangular window has area 24 square feet and height six feet. What is its base?
33. What is the base of a triangle with area 207 square inches and height 18 inches?
 Answer

23 inches
34. What is the height of a triangle with area 893 square inches and base 38 inches?
35. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.
 Answer

\(45°,\; 45°,\; 90°\)
36. The measure of the smallest angle of a right triangle is \(20°\) less than the measure of the next larger angle. Find the measures of all three angles.
37. 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.
 Answer

\(30°,\; 60°,\; 90°\)
38. The angles in a triangle are such that one angle is \(20\) more than the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
39.
 Answer

\(15\)
40.
41.
 Answer

\(25\)
42.
In the following exercises, use the Pythagorean Theorem to find the length of the unknown leg. Round to the nearest tenth if necessary.
43.
 Answer

\(8\)
44.
45.
 Answer

\(12\)
46.
47.
 Answer

\(10.2\)
48.
49.
 Answer

\(9.8\)
50.
In the following exercises, solve using a geometry formula.
51. The width of a rectangle is seven meters less than the length. The perimeter is \(58\) meters. Find the length and width.
 Answer

\(18\) meters, \(11\) meters
52. The length of a rectangle is eight feet more than the width. The perimeter is \(60\) feet. Find the length and width.
53. 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.
 Answer

\(13.5\) m, \(12.8\) m
54. The length of the rectangle is \(1.1\) meters less than the width. The perimeter of a rectangle is \(49.4\) meters. Find the dimensions of the rectangle.
55. 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.
 Answer

\(25\) ft, \(50\) ft
56. The length of the rectangle is three times the width. The perimeter of a rectangle is \(72\) feet. Find the length and width of the rectangle.
57. 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.
 Answer

\(7\) m, \(11\) m
58. The length of a rectangle is five inches more than twice the width. The perimeter is \(34\) inches. Find the length and width.
59. 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.
 Answer

\(12\) ft, \(13\) ft, \(14\) ft
60. The perimeter of a triangle is \(35\) feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.
61. 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.
 Answer

\(3\) ft, \(6\) ft, \(8\) ft
62. One side of a triangle is three times the smallest side. The third side is three feet more than the shortest side. The perimeter is \(13\) feet. Find the lengths of all three sides.
63. 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.
 Answer

\(120\) yd, \(160\) yd
64. The perimeter of a rectangular atrium is \(160\) feet. The length is \(16\) feet more than the width. Find the length and width of the atrium.
65. 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.
 Answer

\(40\) ft, \(85\) ft
66. A rectangular rug has perimeter \(240\) inches. The length is \(12\) inches more than twice the width. Find the length and width of the rug.
In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.
67. 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?
 Answer

\(5\) feet
68. Pam wants to put a banner across her garage door diagonally, as shown, to congratulate her son for his college graduation. The garage door is \(12\) feet high and \(16\) feet wide. How long should the banner be to fit the garage door?
69. 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?
 Answer

\(14.1\) feet
70. Brian borrowed a \(20\)foot extension ladder to use when he paints his house. If he sets the base of the ladder six feet from the house as shown, how far up will the top of the ladder reach?