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 The Pythagorean Theorem In any right triangle, if a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse, then we have the following  or alternately Other types of applications require us to interpret and use a given formula.

Example 3:

Find the missing side of the right triangle if .

Solution:

First we start by drawing the picture So we can see that we are clearly missing side b. Therefore we can use the formula . We get So .

Example 4:

The equation for the time of one pendulum swing (called the period of the pendulum) is given by where T is the time in seconds and L is the length in feet. Find the length of a
pendulum of a clock that has a period of 2.3 seconds.

Solution:

We simply need to put the value of 2.3 into the formula for T and solve for L by using the
techniques we learned in this section. We proceed as follows Now we put this into our calculator to get . Upon checking this value, we find that
it does check and therefore the length of the pendulum is 17.15 feet.

8.7 Exercises

Solve.  Solve for the indicated variable. Find the missing side of the right triangle. 63. A ten foot ladder is leaning against a wall. How far is the bottom of the ladder from the wall
when the ladder reaches a height of 8 feet?

64. A 25 foot ladder is leaning against a building. How high up the building does the ladder
reach if the bottom of the ladder is 5 feet from the building?

65. The diagonal of a TV is 15 inches. The width of the TV is 10 inches. What is the height of
the TV?

66. A computer monitor has a width of 13 inches and a height of 10 inches. What is the length of
the diagonal of the monitor?

67. A stereo receiver is in a corner of a 12 foot by 14 foot room. Speaker wire will run under a
rug, diagonally, to a speaker in the far corner. If a 4 foot slack is needed at each end, how
long of a piece of wire should be used?

68. A baseball diamond is a square which is 90 feet on the side. What is the distance from first base
to third base?
69. The formula for the speed of a falling object is where v is the speed of the object
in feet per second and d is the distance the object has fallen, in feet. What distance has an
object fallen if its speed is 150 feet per second?

70. The formula for the speed of a falling object is where v is the speed of the object
in feet per second and d is the distance the object has fallen, in feet. What distance has an
object fallen if its speed is 75 feet per second?

71. The formula for the distance a lookout can see is where d is the distance in
miles and h is the height of the lookout above the water, in feet. How high will the periscope
have to go to see a ship that is 6.5 miles away?

72. The formula for the distance a lookout can see is where d is the distance in
miles and h is the height of the lookout above the water, in feet. How high will the periscope
have to go to see an island that is 4.3 miles away?

73. The equation for the period of the pendulum is given by where T is the time in
seconds and L is the length in feet. Find the length of a pendulum of a clock that has a
period of 4 seconds.
74. The equation for the period of the pendulum is given by where T is the time in
seconds and L is the length in feet. Find the length of a pendulum that has
a period of 7.2 seconds.

75. The formula for the distance required of a moving object to reach a specific velocity is where v is the velocity of the object, a is the acceleration of the object and s is
the distance required. What distance is required for an car to reach a velocity of 50 miles per
hour if the acceleration is 5 miles per square second?

76. The formula for the distance required of a moving object to reach a specific velocity is where v is the velocity of the object, a  is the acceleration of the object and s is
the distance required. What distance is required for an car to reach a velocity of 72
kilometers per hour if the acceleration is 10 kilometers per square second?

77. The formula for the demand for a certain product is given by where x
is the number of units demanded per day and p is the price per unit. What is the demand if
the price is \$37.55?
78. The formula for the demand for a certain product is given by where x
is the number of units demanded per day and p is the price per unit. What is the demand if
the price is \$34.70?
79. The formula for the escape velocity of a satellite is where v is the velocity,
g is the force of gravity, r is the planets radius and h is the height of the satellite above the
planet. What is the height above the Earth would the Space Shuttle be if the gravitational
pull is 0.0098 km/sec2 and the escape velocity is 2.45 km/sec? (The radius of the earth is
80. The formula for the escape velocity of a satellite is where v is the velocity,