91Ó°ÊÓ

A six-foot-tall person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is 132 feet from the tower and 3 feet from the tip of the shadow, the person's shadow starts to appear beyond the tower's shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the tower?

A sprinkler on a golf green sprays water over a distance of 15 meters and rotates through an angle of \(140^{\circ} .\) Draw a diagram that shows the region that the sprinkler can irrigate. Find the area of the region.

A car's rear windshield wiper rotates \(125^{\circ} .\) The total length of the wiper mechanism is 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper.

In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is \(3.5^{\circ} .\) After you drive 13 miles closer to the mountain, the angle of elevation is \(9^{\circ}\) (see figure). Approximate the height of the mountain.

A 20 -meter line is a tether for a helium-filled balloon. Because of a breeze, the line makes an angle of approximately \(85^{\circ}\) with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write and solve an equation for the height of the balloon. (c) The breeze becomes stronger and the angle the line makes with the ground decreases. How does this affect the triangle you drew in part (a)? (d) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures \(\theta\) $$\begin{array}{|l|l|l|l|l|} \hline \text { Angle, } \boldsymbol{\theta} & 80^{\circ} & 70^{\circ} & 60^{\circ} & 50^{\circ} \\ \hline \text { Height } & & & & \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|} \hline \text { Angle, } \theta & 40^{\circ} & 30^{\circ} & 20^{\circ} & 10^{\circ} \\ \hline \text { Height } & & & & \\ \hline \end{array}$$ (e) As \(\theta\) approaches \(0^{\circ},\) how does this affect the height of the balloon? Draw a right triangle to explain your reasoning.

The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The railway cars travel a distance of 896.5 feet at an angle of approximately \(35.4^{\circ},\) rising to a height of 1693.5 feet above sea level. (a) Find the vertical rise of the inclined plane. (b) Find the elevation of the lower end of the inclined plane. (c) The cars move up the mountain at a rate of 300 feet per minute. Find the rate at which they rise vertically.

In right triangle trigonometry, explain why \(\sin 30^{\circ}=\frac{1}{2}\) regardless of the size of the triangle.

Write an equation for the function that is described by the given characteristics. A sine curve with a period of \(\pi,\) an amplitude of 2 a right phase shift of \(\pi / 2,\) and a vertical translation up 1 unit

Distance A plane flying at an altitude of 7 miles above a radar antenna will pass directly over the radar antenna (see figure). Let \(d\) be the ground distance from the antenna to the point directly under the plane and let \(x\) be the angle of elevation to the plane from the antenna. (d is positive as the plane approaches the antenna.) Write \(d\) as a function of \(x\) and graph the function over the interval \(0 < x < \pi\).

The function $$P=100-20 \cos \frac{5 \pi t}{3}$$ approximates the blood pressure \(P\) (in millimeters of mercury) at time \(t\) (in seconds) for a person of rest. (a) Find the period of the function. (b) Find the number of heartbeats per minute.