Question

question 2
a roller coaster enthusiast wanted to find the exact height of their favorite roller coaster at any given time during the ride. using the following equation
$$h(t) = 4t^3 - 52t^2 + 128t + 420,$$
they predicted the height of the roller coaster, ( h ), in feet, of the ride ( t ) seconds after it started.

interpret the meaning of ( (10, 500) ).

a. they estimated the height of the roller coaster to be about 10 feet after 500 seconds.
b. they estimated the height of the roller coaster to be about 500 feet after 10 seconds.
c. they estimated the height of the roller coaster to be increasing by 500 feet every 10 seconds.
d. they estimated the height of the roller coaster to be increasing by 10 feet every 500 seconds.

Explanation:

In the function \( H(t) = 4t^3 - 52t^2 + 128t + 420 \), \( t \) represents the time in seconds after the ride starts, and \( H(t) \) represents the height in feet. A point \((10, 500)\) means when \( t = 10 \) (10 seconds after the ride starts), the height \( H(10) = 500 \) feet.

• Option A swaps time and height, so it's incorrect.
• Option C and D refer to the rate of change (slope), but \((10, 500)\) is a point, not a rate, so they are incorrect.
• Option B correctly interprets \( t = 10 \) seconds and \( H = 500 \) feet.

Answer:

B. They estimated the height of the roller coaster to be about 500 feet after 10 seconds.