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A diving board is 10 feet above the surface of the water. At time t = 0, the board propels Chris upward at a rate of 12 feet per second and he enters the water feet first. The equation h = –16t^2 + 12t + 10 can be used to find Chris's height after t seconds. How long will it take Chris to return to his starting height of 10 feet?

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Answer:

Chris will take 0.75 seconds to return to his starting height of 10 feet.

Step-by-step explanation:

Let the height of Chris be represented by [tex]h(t) = -16\cdot t^{2}+12\cdot t +10[/tex], where [tex]h(t)[/tex] is the height in feet and [tex]t[/tex], the time in seconds. First, we equalize the formula to a height of 10 feet and simplify the resulting expression, that is:

[tex]-16\cdot t^{2}+12\cdot t +10 = 10[/tex]

[tex]-16\cdot t^{2}+12\cdot t = 0[/tex]

Then, we simplify the expression by algebraic means:

[tex]-16\cdot t\cdot \left(t-\frac{3}{4} \right) = 0[/tex]

Roots of the polynomial are, respectively:

[tex]t = 0\,s\,\lor \,t = 0.75\,s[/tex]

First root represents the initial height of Chris, whereas the second one represents the instant when Christ returns to the same height above the surface of the water. Hence, Chris will take 0.75 seconds to return to his starting height of 10 feet.

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