Question

1. a slide 4.1 meters long makes an angle of elevation of 35° with the ground. to the nearest tenth of a meter, how far above the ground is the top of the slide?

find the value of x. round the length to the nearest tenth.

not drawn to scale

17.

not drawn to scale

18.

1. a tree that is 10 yards tall casts a shadow that is 14 yards long. find the angle of elevation of the sun ground. draw a picture. label the sides. write an equation and solve. round to the nearest degree.

Explanation:

Step1: Use sine function

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For the first problem (assuming it's related to the first triangle with hypotenuse \(4.1\)m and angle \(35^{\circ}\)), if \(x\) is the height (opposite side), then \(\sin(35^{\circ})=\frac{x}{4.1}\). So \(x = 4.1\times\sin(35^{\circ})\). Using a calculator, \(\sin(35^{\circ})\approx0.5736\), and \(x\approx4.1\times0.5736 = 2.35176\approx2.4\)m.

Step2: For the second triangle (hypotenuse \(15\)m, angle \(35^{\circ}\), opposite side \(x\))

Using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\sin(35^{\circ})=\frac{x}{15}\). Then \(x = 15\times\sin(35^{\circ})\). Since \(\sin(35^{\circ})\approx0.5736\), \(x\approx15\times0.5736=8.604\approx8.6\)m.

Step3: For the third triangle (opposite side \(200\)m, angle \(10^{\circ}\), hypotenuse \(x\))

Using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\sin(10^{\circ})=\frac{200}{x}\), so \(x=\frac{200}{\sin(10^{\circ})}\). Since \(\sin(10^{\circ})\approx0.1736\), \(x=\frac{200}{0.1736}\approx1151.8\)m.

Step4: For the tree problem (opposite side \(10\) yards, adjacent side \(14\) yards)

Using \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), \(\tan\theta=\frac{10}{14}\approx0.7143\). Then \(\theta=\arctan(0.7143)\). Using a calculator, \(\theta\approx35.54^{\circ}\approx36^{\circ}\).

Answer:

1. d. \(2.4\)m
2. c. \(8.6\)m
3. b. \(1151.8\)m
4. b. \(36^{\circ}\)