Question

While hiking, Marek throws a rock off a cliff that is roughly 10 meters above a lake. This situation is modeled by the equation h=−4.9t^2 +10 , with h representing the height in meters and t the time in seconds. If you were to set up a table to determine when the rock hits the water, what would be a reasonable scale for the range?

a. 5
b. 1/10
c. 2
d. 10

Answer 1

To determine when the rock hits the water, we need to find the time \( t \) when the height \( h \) is zero (i.e., when the rock reaches the lake's surface).

The equation given is:

\[ h = -4.9t^2 + 10 \]

To find when the rock hits the water, we set \( h = 0 \):

\[ 0 = -4.9t^2 + 10 \]

Rearranging gives:

\[ 4.9t^2 = 10 \]

Dividing both sides by 4.9:

\[ t^2 = \frac{10}{4.9} \approx 2.0408 \]

Taking the square root:

\[ t \approx \sqrt{2.0408} \approx 1.43 \, seconds \]

Now, we can consider reasonable scales for the range of \( t \). Given that we want to generate a table for \( t \), we should include values around 0 and up to around 1.5 seconds.

Let's evaluate the options for the scale:

- a. **5** - This would result in a very large range unnecessarily since values like 5 seconds are far beyond when the rock will hit the water.
- b. **1/10** - This scale (0.1) would provide very detailed increments, allowing us to capture the moment the rock hits the water accurately.
- c. **2** - This could fit, but 2 seconds is still larger than the actual time we calculated (1.43 seconds).
- d. **10** - Similar to option A, this scale is much too large; we won't need such a range of time.

Given these evaluations, **the best reasonable scale for the range is b. 1/10**, allowing for a detailed view of the rock's descent and the time it hits the water.