Question

ROLLER COASTERS On a racing roller coaster, two trains start at the same time and race to see which returns to the station first. On one coaster, the height of a train on the blue track can be modeled by f(x)=1/20(x^3−60x^2+900x)
and the height of a train on the green track can be modeled by g(x)=1/12.000(x^5−144x^4+7384x^3−158,400x^2+1,210,000x)
where x is time in seconds for the first 35 seconds of the ride.

a. Write an equation to determine the times when the blue and green trains are at the same height. Then, write an equation to determine the times for which the blue train modeled by f(x) is at a height of 150 feet. Separate the two equations with a semicolon.

Part B
Fill in the blank question.
Use a graphing calculator to sketch a graph and solve each equation. List solutions in order from least to greatest and rounded to the nearest tenth.

At what times are the blue and green trains at the same height?

At what times is the blue train at a height of 150 feet?

Answer 1

Answer: To find when the blue and green trains are at the same height, set their respective height functions equal to each other. To determine when the blue train reaches a height of 150 feet, set the function of the blue train's height equal to 150 and solve for time.

Explanation:

To determine the times when the blue and green trains of the roller coaster are at the same height, we need to set their equations equal to each other:

f(x) = g(x)

So, 1/20 (x³ - 60x² + 900x) = 1/12.000 (x⁵ - 144x⁴ + 7384x³ -158,400x² +1,210,000x).

This equation can be solved for x to find the specific times when both trains are at the same height on their respective tracks. As for the equation to determine the times when the blue train is at a height of 150 feet, we set its height equation equal to 150:

f(x) = 150

So, 1/20 (x³ - 60x² + 900x) = 150.

The solutions of this equation will give the times at which the blue train reaches 150 feet.