Question
On the moon, the equation \( d=0.8 t^{2} \) is used to find the distance an object has fallen after \( t \) seconds. On Earth, the equation is \( d=4.9 t^{2} \). A rock is thrown from a height of 60 meters on Earth. Plot the graph of the equation on a graphing calculator or other technology to find the time taken to the ground to the nearest second. \( t= \) Enter your next step here s
Ask by Clarke Schwartz. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( t = 4 \) seconds
Solution
To determine the time it takes for the rock to reach the ground, follow these steps:
1. **Set Up the Equation:**
The distance \( d \) fallen is given by the equation:
\[
d = 4.9 t^{2}
\]
Since the rock is thrown from a height of 60 meters, set \( d = 60 \) meters:
\[
60 = 4.9 t^{2}
\]
2. **Solve for \( t \):**
Divide both sides by 4.9 to isolate \( t^{2} \):
\[
t^{2} = \frac{60}{4.9} \approx 12.2449
\]
Take the square root of both sides to find \( t \):
\[
t \approx \sqrt{12.2449} \approx 3.5 \text{ seconds}
\]
3. **Round to the Nearest Second:**
Rounding 3.5 seconds to the nearest whole number gives:
\[
t \approx 4 \text{ seconds}
\]
**Answer:**
\( t = 4 \) seconds
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Simplify this solution Beyond the Answer
To find the time taken for the rock to hit the ground when thrown from a height of 60 meters, we start with the equation \(d = 60 - 4.9t^{2}\). We want to determine when the rock reaches the ground, which is when the distance \(d\) equals 0. Set the equation to zero: \[ 0 = 60 - 4.9t^{2}. \] Rearranging gives us: \[ 4.9t^{2} = 60 \] Now, divide both sides by 4.9: \[ t^{2} = \frac{60}{4.9} \] Taking the square root will yield: \[ t \approx \sqrt{12.2449}, \] which gives approximately \( t \approx 3.49 \) seconds. Rounding this to the nearest second, we find \( t \approx 3 \) seconds. So the time taken to hit the ground is \( t = 3 \) s.
