Throwing a stone Suppose a stone is thrown vertically upward f...

Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.
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 diver jumps off a diving board 10 ft above the pool with an initial upward velocity of 16 ft per second. The height and feet of the diver above the water t seconds after the jump is given by S of T equals -16 T2 + 16 T + 10. What is the maximum height of the diver above the water? A 24 ft, B 16 ft, C 14 ft, and D 10 ft. So we're given the height functions of T. And we want to identify its maximum, so we have to take derivative of S of T. This means that we're differentiating negative 16 T2 plus 16 T + 10. So here we can apply the power rule which gives us -302 T. Plus 16 + 0, the derivative of a constant is 0, and we want to set it to 0. To identify the critical points. This means that -302 T. Plus 16 is equal to 0. And therefore, if we rearrange, we get 16 equals 32 T. Dividing both sides by 302, we get T equals 16 divided by 32. Which is 1/2. So this is a critical point, and now we can check the sign of the derivative before it and after. So, let's identify as prime for T less than 1/2. We can take time of 0, so let's take S at 0.0. This is equal to -32 multiplied by 0 + 16, which is 16 and this is positive. Now let's identify as prime for time values greater than 1/2, such as 1. So we got S at 1, and this gives us -32 + 16, which is -16, and this is less than 0. So the derivative changes signed from positive to negative, meaning our function was increasing and started to decrease. So this is the point of maximum. So time of 1.5 seconds is the point of maximum, and now that we have proved that this is the point of maximum, we can identify the maximum height by evaluating S at 1/2. So we are evaluating the height function at the point of maximum. We get -16 multiplied by 1/2 squared plus 16. Multiplied by 1/2 plus 10. And now we have to simplify so we get -16 multiplied by 1/4. Plus 16 divided by 2, this gives us 8, and then we are adding 10. So overall, we get -4. Plus 18 Which is equal to 14. The units are given in feet, so the correct answer to this problem corresponds to the answer choice C 14 ft. Thank you for watching.

Key Concepts

Quadratic Functions

Vertex of a Parabola

Maximizing Functions

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