Suppose a stone is thrown vertically upward from the edge of a...

Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t²+19.6t+24.5.
c. What is the height of the stone at the highest point?

c. What is the height of the stone at the highest point?

Accepted Answer

Welcome back, everyone. A child throws a toy airplane vertically upward from a balcony 10 m above the ground with an initial speed of 9.8 m per second. The height and meters of the airplane above the ground 10 seconds after it thrown is given by A of T equals -4.9 T2 plus 9.8 T + 10. What is the maximum height the toy airplane reaches were given four answer choices A says 14.9, B 11.9, C 19.4, and D 41.9, all given in meters. So we are given our height function A of T equals -4.9 T squared plus 9.8 T + 10. And we want to find its maximum. So what we have to do is identify its first derivative, which gives us the velocity function, right? And essentially we're going to apply the power rule because we're differentiating -4.9 T2. Plus 9.8 T plus 10. So let's identify the derivative using the power rule. The derivative of T squared is 2T. So we get 9.8T. The derivative of T is 1, so we get 9.8, and the derivative of 10 is 0. And we're going to set it equal to 0 to identify the critical points. So we set it equal to 0, which means that -9.8 T plus 9.8 is equal to 0, and we can essentially solve it and see that time is equal to 1, right? We can just add 9.8 T to both sides and divide both sides by 9.8, which gives us t of 1. So now we want to check if this point is a point of minima or maxima. What we can do is simply evaluate the derivative at time being less than one. And then we are going to evaluate it at time being greater than 1. So for the first one, we can just take the derivative at 0. Which is in the domain, right? And specifically, we are going to get -9.8 multiplied by 0 plus 9.8. So that's 9.8, and this is greater than 0, which means that the function is increasing. And then we are going to take time value, which is greater than 1, let's suppose that we take 2. So A at 2 is equal to -9.8 multiplied by 2 + 9.8. So we get -19. 0.6 + 9.8. This is less than 0, right? Because the negative term. Has a larger magnitude than the positive term, right, so it's negative, and this means that the function starts to decrease. So if our function was increasing and started to decrease after the critical point of 1, this means that we have a local maximum. So we have proved that this point corresponds to the local maximum and that we want to identify the height. So let's value it A at 0.1, which gives us -4.9 multiplied by 1 squared. 9.8 multiplied by 1 + 10. So we get -4.9 plus 9.8, which is 4.9 plus 10. Overall, that's 14.9. And height is measured in meters in this context, so our final answer is option A, 14.9 m. Thank you for watching.

Key Concepts

Quadratic Functions

Vertex of a Parabola

Maximum Value of a Function

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