Math

Question Find the time (in seconds) when Ana, who dives from a high springboard, hits the water given the height model h(x)=5(x+1)(x3)h(x) = -5(x+1)(x-3).

Studdy Solution

STEP 1

Assumptions
1. The height function h(x)h(x) models Ana's height above the water in meters.
2. The variable xx represents the time in seconds after diving.
3. Ana hits the water when her height above the water is 0 meters.

STEP 2

To find the time when Ana hits the water, we need to solve for xx when the height h(x)h(x) is equal to 0.
h(x)=0h(x) = 0

STEP 3

Now, plug in the given height function into the equation.
5(x+1)(x3)=0-5(x+1)(x-3) = 0

STEP 4

To solve for xx, we can use the zero product property, which states that if a product of factors is zero, then at least one of the factors must be zero.
5(x+1)(x3)=0-5(x+1)(x-3) = 0

STEP 5

We can ignore the constant factor 5-5 because it does not affect the solution for xx.
(x+1)(x3)=0(x+1)(x-3) = 0

STEP 6

Set each factor equal to zero and solve for xx.
x+1=0orx3=0x+1 = 0 \quad \text{or} \quad x-3 = 0

STEP 7

Solve the first equation for xx.
x+1=0x+1 = 0 x=1x = -1

STEP 8

Solve the second equation for xx.
x3=0x-3 = 0 x=3x = 3

STEP 9

Since time cannot be negative, we discard the negative solution x=1x = -1.

STEP 10

The positive solution x=3x = 3 represents the time in seconds after diving when Ana hits the water.
Ana will hit the water 3 seconds after diving.

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