# A diver jumps from a platform 25 feet above the surface of the water. The diver's height above the water is given by the equation h(t)=-16t^2+14.5t + 25, where t is the time, in seconds, after the diver jumps.

A.)When will the diver reach a height of 32 feet?
B.)How long will it take the diver to reach the water?

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## A. solve -16t^2+14.5t + 25 = 32

B. solve -16t^2+14.5t + 25 = 0

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## To find when the diver reaches a height of 32 feet, we will set h(t) equal to 32 and solve for t.

A.) h(t) = 32

-16t^2 + 14.5t + 25 = 32

-16t^2 + 14.5t + 25 - 32 = 0

-16t^2 + 14.5t - 7 = 0

Using the quadratic formula, where a = -16, b = 14.5, and c = -7:

t = (-14.5 ± √(14.5^2 - 4(-16)(-7))) / (2(-16))

t = (-14.5 ± √(210.25 - 448)) / (-32)

t = (-14.5 ± √(210.25 + 448)) / (-32)

Now, calculate the value under the square root:

t = (-14.5 ± √658.25) / (-32)

t = (-14.5 ± 25.65) / (-32)

Now, solve for the two possible values of t:

t = (-14.5 + 25.65) / (-32)

t = 11.15 / (-32)

t ≈ -0.35

t = (-14.5 - 25.65) / (-32)

t = -40.15 / (-32)

t ≈ 1.26

Therefore, the diver will reach a height of 32 feet at approximately 1.26 seconds after jumping.

B.) To find how long it takes for the diver to reach the water, we need to find the time when the height is 0.

Setting h(t) = 0:

-16t^2 + 14.5t + 25 = 0

t = (-14.5 ± √(14.5^2 - 4(-16)(25))) / (2(-16))

t = (-14.5 ± √(210.25 + 1600)) / (-32)

t = (-14.5 ± √(1810.25)) / (-32)

Now, solve for the two possible values of t:

t = (-14.5 + √1810.25) / (-32)

t ≈ 3.99

t = (-14.5 - √1810.25) / (-32)

t ≈ -0.24

Therefore, it will take approximately 3.99 seconds for the diver to reach the water.

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## To find the time when the diver reaches a height of 32 feet, we need to set the equation for the height equal to 32 feet and solve for t.

So, we have:
-16t^2 + 14.5t + 25 = 32

We can simplify the equation by moving 32 to the other side:
-16t^2 + 14.5t + 25 - 32 = 0

Simplifying further:
-16t^2 + 14.5t - 7 = 0

t = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = -16, b = 14.5, and c = -7.

Substituting these values into the quadratic formula, we get:
t = (-14.5 ± √(14.5^2 - 4(-16)(-7))) / (2(-16))

After simplification, we have:
t = (-14.5 ± √(210.25 - 448)) / (-32)

Further simplifying:
t = (-14.5 ± √(-237.75)) / (-32)

Since we have a negative value inside the square root, there are no real solutions for t. Therefore, the diver does not reach a height of 32 feet.

To find the time it takes for the diver to reach the water, we need to find the time when the height is 0. This is because the water's surface is at a height of 0 feet.

So, we set h(t) = 0 and solve for t:
-16t^2 + 14.5t + 25 = 0

Using the quadratic formula, we get:
t = (-14.5 ± √(14.5^2 - 4(-16)(25))) / (2(-16))

Simplifying further:
t = (-14.5 ± √(210.25 + 1600)) / (-32)

t = (-14.5 ± √(1810.25)) / (-32)

After calculating the square root and simplifying, we find:
t ≈ -0.48 seconds or t ≈ 1.90 seconds

Since time cannot be negative, the diver reaches the water approximately 1.90 seconds after jumping from the platform.