Whats the maximum height the diver will reach?

Question

Suppose h(t)=-5t²+10t+3 is an expression giving the height of a diver above the water in meters, t seconds after the diver leaves the springboard. Answer the following questions

a.the height of the springboard above the water?

b. the amount of time it would take the diver to hit the water?

c. the time on the divers descent toward the water that the diver is the same height as the springboard?

d. the time it takes the diver to reach the peak in their dive? 

e. the maximum height the diver will reach?

Answer 1

a) The height of the springboard above the water should be h(0) : Read, the height at t = 0

 

h(0) = -5(0) + 10(0) + 3

h(0) = 0 + 0 + 3

h(0) = 3

 

a) 3 meters

b)  The time it takes the diver to hit water should be, the positive 0 solution for t.  Remember, in a quadratic equation,  there are two values for t where a parabola crosses the horizontal axis, which in this case would be t.  Just by looking at the function, h(t) = -5t² + 10t + 3 , one should be able to see that it cannot be factored easily, so it requires the Quadratic Formula to find the zeros ; x = -b ±√(b²-4ac)  / 2a

 

Substitute t for x, and use the coefficients for a, b, c:

 

t  =  (-10 ± √((10² - 4(-5)(3)))/2(-5)

t = (-10 ±√(100 + 60))/-10

t = (-10 ±√160)/-10) ; Now factor the 160 to simplify:

t = (-10 ±√(10*16))/-10

t = (-10 ±4√10)/-10 ; Factor out leading coefficient of -2 from the numerator:

t = -2(5 ± 2√10)/-10

t = (5 ± 2√10)/5

Using a calculator to find the zeros, and disregarding the negative zero (because t starts at 0):

t ≈ 2.265

b) approx. 2.265 seconds for diver to hit water.

c) To find this, set the function equal to 3 to find what other value for t would be equal to 3 (we know one is 0).

 

-5t^t + 10t + 3 = 3

-5t² + 10t = 0 ; factor out t

t(-5t + 10) = 0

We know t = 0:

We also know that -5t + 10 = 0

-5t = -10

t = 2

c) 2 seconds. This is the time that diver would equal height of t=0 which is where he started, and where he equals the height of the springboard.

 

d and e) The peak of the dive (parabola), is determined using the formula h = -b/2a (Derived from the Quadratic Formula) to find the y value (in this case, the h value, answering e) and then using that result in the function to find the x value (in this case, the t value answering d) of the point where the parabola (dive path) reaches a maximum(height), or minimum(in upward opening parabolas).

 

h = -10/2(-5)

h = -10/-10

h = 1

h(1) = -5(1)² + 10(1) + 3

h(1) = -5 + 10 + 3

h(1) = 8

 

d) At t = 1 second, diver will have reached peak of dive.

e) At t = 1 second, diver will have reached a maximum height of 8 meters.