Final answer:
a) The rollercoaster will take approximately 4.1 seconds to rise 250 feet. b) The velocity of the rollercoaster when it is 250 feet above is 80 feet per second. The acceleration is 20 feet per second squared.
Step-by-step explanation:
a) To find the time it will take for the rollercoaster to rise 250 feet, we can set up the equation s(t) = 250 and solve for t. The equation is given as s(t) = 10t² - 4t + 20. So, 10t² - 4t + 20 = 250. This can be rewritten as 10t² - 4t - 230 = 0. Using the quadratic formula, we can find that t ≈ 4.1 seconds.
b) To find the velocity when the rollercoaster is 250 feet above the ground, we need to find the derivative of the function s(t) with respect to t. The derivative is given as v(t) = 20t - 4. When the rollercoaster is 250 feet above, we can substitute s(t) = 250 into the derivative to find v(t). So, v(t) = 20t - 4 = 20(4.1) - 4 = 80. Therefore, the velocity of the rollercoaster when it is 250 feet above is 80 feet per second.
To find the acceleration, we take the derivative of the velocity function. The derivative of v(t) = 20t - 4 is a(t) = 20. Therefore, the acceleration of the rollercoaster when it is 250 feet above is 20 feet per second squared.