Question

Fatima and Tia are loading a refrigerator into the back of a truck. They know they need the ramp to make a 50-degree angle with the ground to reach the truck bed, which is 42 inches from the ground. How long is the ramp? Option #1: sin 50°=42x=55 inches Option #2: cos 50°=42x=48 inches Option #3: sin 42°=x42=63 inches Option #4: cos 42°=42x=62 inches (1 point) The length of the ramp is Option #$$.

Answer 1

To find the length of the ramp, we can use trigonometric functions. The ramp forms a right triangle with the vertical height (42 inches) as the opposite side and the length of the ramp as the hypotenuse. The angle of the ramp is 50 degrees.

We'll use the sine function, which relates the opposite side to the hypotenuse as follows:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case, we have:

\[
\sin(50^\circ) = \frac{42}{\text{hypotenuse}}
\]

Let \( x \) be the length of the ramp (hypotenuse). Therefore:

\[
\sin(50^\circ) = \frac{42}{x}
\]

Now we can rearrange the equation to solve for \( x \):

\[
x = \frac{42}{\sin(50^\circ)}
\]

Now, calculate \( \sin(50^\circ) \):

\[
\sin(50^\circ) \approx 0.7660
\]

Substituting this value into the equation gives us:

\[
x \approx \frac{42}{0.7660} \approx 54.8 \text{ inches}
\]

So, the closest option in the context of the choices provided is:

Option #1: \( \sin 50^\circ = \frac{42}{x} \Rightarrow x \approx 55 \text{ inches} \).

Thus, the length of the ramp is \( \boxed{55} \) inches.