Question

1. a 14-ft-long ramp rises at an angle of 22.2°. how long is the base of the ramp to the nearest foot?

f 11 ft
g 13 ft
h 17 ft
i 22 ft

1. what is the value of w to the nearest degree?

a 25
b 35
c 40
d 45
(there is a right triangle diagram with one leg 15, hypotenuse 21, one angle 65°, right angle, and angle w°)

Explanation:

Question 4

Step1: Identify the triangle type

The ramp forms a right triangle, with the length of the ramp as the hypotenuse (\(c = 14\) ft) and the angle of elevation \(\theta = 22.2^\circ\). We need to find the base (adjacent side, \(b\)) of the right triangle.

Step2: Use the cosine function

In a right triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(22.2^\circ)=\frac{b}{14}\).

Step3: Solve for \(b\)

Multiply both sides by 14: \(b = 14\times\cos(22.2^\circ)\). Calculate \(\cos(22.2^\circ)\approx0.926\), then \(b\approx14\times0.926 = 12.964\approx13\) ft.

Step1: Analyze the right triangles

First, consider the left right triangle with angle \(65^\circ\) and hypotenuse 15. We can find the height \(v\) using \(\sin(65^\circ)=\frac{v}{15}\), so \(v = 15\times\sin(65^\circ)\approx15\times0.9063 = 13.5945\).

Step2: Use the right triangle with hypotenuse 21 and height \(v\)

Now, in the right triangle with hypotenuse 21 and height \(v\), we can find the angle \(w\) using \(\sin(w)=\frac{v}{21}\). Substitute \(v\approx13.5945\), so \(\sin(w)=\frac{13.5945}{21}\approx0.6474\). Then \(w=\arcsin(0.6474)\approx40^\circ\).

Answer:

G. 13 ft

Question 5