Question

The distance from the Grand canyon decreases 150 mi every 3h, After 4h the distance from the grand canyon is 200 mi. the rate of change is -50,What Is the initial value?

Answer 1

To find the initial value, we can model the distance from the Grand Canyon as a linear function based on the information given.

First, we know the following:
- The distance decreases by 150 miles every 3 hours. This means the rate of change is \(-50\) miles per hour (since \(150 \div 3 = 50\), and it's decreasing).
- After 4 hours, the distance from the Grand Canyon is 200 miles.

We can denote the distance from the Grand Canyon as \(d(t)\), where \(t\) is the time in hours.

Starting with the general linear equation of the form:
\[
d(t) = mt + b
\]
where:
- \(m\) is the rate of change (\(-50\) miles per hour),
- \(b\) is the initial value (distance at \(t = 0\)).

Given that after 4 hours (\(t = 4\)), the distance is 200 miles, we can set up the equation:
\[
d(4) = -50(4) + b = 200
\]

Now, solve for \(b\):
\[
-200 + b = 200
\]
\[
b = 200 + 200
\]
\[
b = 400
\]

Thus, the initial value (the distance from the Grand Canyon at \(t = 0\)) is **400 miles**.