Suppose the position of an object moving horizontally after se...

Question

Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?

Accepted Answer

Welcome back, everyone. A projectile is launched vertically upwards and its height above the ground after T seconds is given by the function H of T equals 50 T minus 5 T squad, where T is between 0 and 10 inclusive. At what time is the projectile at its maximum height? We given for answer choices a. Time equals 5 seconds, B 10 seconds. C 15 seconds, and D 20 seconds. So we want to maximize the function and we can do that by identifying the derivative H of T, which means that we're taking the derivative of. 5 TT and subtracting the derivative of 5 t squad. We can apply the power rule. The derivative of T is 1, so we get 50, the derivative of T squared is 2T, so we get 50 minus 10 T. And what we want to do is simply set this derivative equal to 0 to identify the critical points, right? So we can factor it out, right, and we're going to get 10 and 5 minus T equals 0. Which essentially means that 5 minus t must be equal to 0. We can divide both sides by 10, right? And solving, we get 5 equals t. So this is our critical point. We want to prove that this point is a point of maximum, and we can prove that by finding the second derivative. So let's differentiate. 50 minus 10 T. Which gives us 0, the derivative of 50 is 0, and the derivative of T is 1, so we get negative 10, and this is less than 0 for any T, which essentially means that the function is concave down. It's indeed a parabola that opens down, so we indeed have a maximum at its vertex, and our final answer is time equals 5 seconds, right? So the correct answer corresponds to the answer choice a. Thank you for watching.

Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?

Key Concepts

Position Function

Critical Points

Second Derivative Test

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