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5. A ramp is set up from a truck's trunk to the ground. The ramp's end at the edge of the trunk is [tex][tex]$4.5 \, \text{ft}$[/tex][/tex] high. If the angle between the ramp and the ground is 16 degrees, find the length of the ramp.

Round your answer to two decimal places if needed.

The length of the ramp is [tex]\square \, \text{ft}[/tex].

Sagot :

GinnyAnswer
To find the length of the ramp given that the height of the truck's trunk above the ground is 4.5 feet and the angle between the ramp and the ground is 16 degrees, we can use trigonometry. Here's how you can solve the problem step-by-step:

1. Identify the given information:
- Height of the trunk above the ground (opposite side of the right triangle): [tex]\( 4.5 \)[/tex] feet.
- Angle between the ramp and the ground: [tex]\( 16 \)[/tex] degrees.

2. Understand the relationship in the right triangle:
We need to find the hypotenuse (the length of the ramp) given the opposite side and the angle. The sine function relates the angle to the opposite side and the hypotenuse:
[tex]\[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]

3. Rearrange the formula to solve for the hypotenuse:
[tex]\[ \text{hypotenuse} = \frac{\text{opposite}}{\sin(\text{angle})} \][/tex]

4. Convert the angle from degrees to radians:
Trigonometric functions in most mathematical contexts use radians. The conversion from degrees to radians is:
[tex]\[ \text{angle in radians} = \text{angle in degrees} \times \frac{\pi}{180} \][/tex]
So, for [tex]\( 16 \)[/tex] degrees:
[tex]\[ \text{angle in radians} = 16 \times \frac{\pi}{180} \approx 0.2792526803190927 \text{ radians} \][/tex]

5. Calculate the sine of the angle in radians:
[tex]\[ \sin(0.2792526803190927) \approx 0.275637355817 \][/tex]

6. Use the sine function to calculate the hypotenuse (length of the ramp):
[tex]\[ \text{hypotenuse} = \frac{4.5}{\sin(0.2792526803190927)} \approx \frac{4.5}{0.275637355817} \approx 16.32579875344485 \text{ feet} \][/tex]

7. Round the answer to two decimal places:
[tex]\[ 16.33 \text{ feet} \][/tex]

Therefore, the length of the ramp is [tex]\( 16.33 \)[/tex] feet.

The length of the ramp is [tex]$\boxed{16.33}$[/tex] feet.
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