Question
A skateboarder is skating at 20 mph when they go down a ramp at a 50° angle below horizontal. If the ramp is 0.04 miles tall, how long will it take them to reach the bottom?
Answers
GPT-4o mini
To determine how long it will take the skateboarder to reach the bottom of the ramp, we can begin by analyzing the situation based on the height of the ramp and the skateboarder's initial speed.
### Step 1: Determine the vertical and horizontal distance covered.
The height of the ramp is given as 0.04 miles. To find the corresponding horizontal distance when the skateboarder descends at a \(50^\circ\) angle, we can use trigonometry (specifically, the sine function since we know the height and angle).
If \( h = 0.04 \) miles, then the total length \( L \) of the ramp can be computed as follows:
\[
h = L \cdot \sin(50^\circ)
\]
So,
\[
L = \frac{h}{\sin(50^\circ)}
\]
Calculating \( L \):
\[
L = \frac{0.04}{\sin(50^\circ)} \approx \frac{0.04}{0.7660} \approx 0.0522 \text{ miles}
\]
### Step 2: Find the time to reach the bottom with initial horizontal speed.
The skateboarder's horizontal speed is \( 20 \text{ mph} \). We can calculate the time \( t \) it takes to cover the horizontal distance \( D \).
First, convert the horizontal distance into the same unit of time (hours):
\[
D = L \cdot \cos(50^\circ) = 0.0522 \cdot \cos(50^\circ) \approx 0.0522 \cdot 0.6428 \approx 0.0335 \text{ miles}
\]
Now, the time \( t \) to reach the bottom is:
\[
t = \frac{D}{\text{horizontal speed}} = \frac{0.0335 \text{ miles}}{20 \text{ mph}}
\]
Calculating \( t \):
\[
t \approx \frac{0.0335}{20} = 0.001675 \text{ hours}
\]
### Step 3: Convert time into seconds.
To convert hours into seconds:
\[
t \approx 0.001675 \text{ hours} \times 3600 \text{ seconds/hour} \approx 6.03 \text{ seconds}
\]
Thus, it will take the skateboarder approximately **6.03 seconds** to reach the bottom of the ramp.
### Step 1: Determine the vertical and horizontal distance covered.
The height of the ramp is given as 0.04 miles. To find the corresponding horizontal distance when the skateboarder descends at a \(50^\circ\) angle, we can use trigonometry (specifically, the sine function since we know the height and angle).
If \( h = 0.04 \) miles, then the total length \( L \) of the ramp can be computed as follows:
\[
h = L \cdot \sin(50^\circ)
\]
So,
\[
L = \frac{h}{\sin(50^\circ)}
\]
Calculating \( L \):
\[
L = \frac{0.04}{\sin(50^\circ)} \approx \frac{0.04}{0.7660} \approx 0.0522 \text{ miles}
\]
### Step 2: Find the time to reach the bottom with initial horizontal speed.
The skateboarder's horizontal speed is \( 20 \text{ mph} \). We can calculate the time \( t \) it takes to cover the horizontal distance \( D \).
First, convert the horizontal distance into the same unit of time (hours):
\[
D = L \cdot \cos(50^\circ) = 0.0522 \cdot \cos(50^\circ) \approx 0.0522 \cdot 0.6428 \approx 0.0335 \text{ miles}
\]
Now, the time \( t \) to reach the bottom is:
\[
t = \frac{D}{\text{horizontal speed}} = \frac{0.0335 \text{ miles}}{20 \text{ mph}}
\]
Calculating \( t \):
\[
t \approx \frac{0.0335}{20} = 0.001675 \text{ hours}
\]
### Step 3: Convert time into seconds.
To convert hours into seconds:
\[
t \approx 0.001675 \text{ hours} \times 3600 \text{ seconds/hour} \approx 6.03 \text{ seconds}
\]
Thus, it will take the skateboarder approximately **6.03 seconds** to reach the bottom of the ramp.