If a rock is propelled upward from the surface of Mars at a velocity of 93 m/sec, it reaches a height

s = 93t – 1.86t^2 after t sec.
a. Find the rock’s velocity and acceleration at time t.

v'(t)=93-3.72t
a''(t)=-3.73

b. How long does it take the rock to reach its highest point?

v'(t)= 93-3.72t=0

t=25s
c. How high does the rock go?

f(x)= 93(25)-1.86(25)^2=1162.5m

I couldn't find the last two.
d. How long does it take the rock to reach 1/4 of its maximum height?
e. How long is the rock aloft?

To find the answers to parts d and e of the given question, let's go step by step.

d. How long does it take the rock to reach 1/4 of its maximum height?
To find the time it takes for the rock to reach 1/4 of its maximum height, we need to set up an equation and solve for t.

Given the equation for the height of the rock: s = 93t - 1.86t^2
Since we are looking for the time it takes for the rock to reach 1/4 of its maximum height, we can set up the following equation:
1/4 of maximum height = s/4 = (93t - 1.86t^2)/4

Now, let's solve for t by setting s/4 equal to the equation for 1/4 of maximum height and then solving the resulting quadratic equation:

s/4 = (93t - 1.86t^2)/4
93t - 1.86t^2 = s
1.86t^2 - 93t + s = 0

Now you can use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1.86, b = -93, and c = s. Substitute these values into the quadratic formula and solve for t.

e. How long is the rock aloft?
To find the total time the rock is aloft, we need to find the time it takes for the rock to reach its maximum height and then double that time.

From part b, we found that the rock reaches its maximum height at t = 25 seconds.

Therefore, the total time aloft = 2 * 25 = 50 seconds.

To summarize:
d. To find how long it takes the rock to reach 1/4 of its maximum height, use the quadratic formula to solve the equation 1.86t^2 - 93t + s = 0.
e. To find how long the rock is aloft, double the time it takes to reach its maximum height.