Solved on Jan 23, 2024

Find the time $t$ when the ball reaches the ground, given the height function $H = -5(t - 3)(t + 5)$ .

STEP 1

Assumptions
1. The equation $H = -5(t - 3)(t + 5)$ represents the height of a ball over time.
2. The variable $t$ represents time.
3. The ball reaches the ground when the height $H$ is equal to $0$ .

STEP 2

To find when the ball reaches the ground, we need to solve for $t$ when $H = 0$ . We set the equation equal to zero and solve for $t$ .
$0 = -5(t - 3)(t + 5)$

STEP 3

Since the product of the terms is equal to zero, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. We set each factor equal to zero and solve for $t$ .

STEP 4

First, set the first factor equal to zero and solve for $t$ .
$(t - 3) = 0$

STEP 5

Add $3$ to both sides of the equation to solve for $t$ .
$t = 3$

STEP 6

Now, set the second factor equal to zero and solve for $t$ .
$(t + 5) = 0$

STEP 7

Subtract $5$ from both sides of the equation to solve for $t$ .
$t = -5$

STEP 8

We have two solutions for $t$ : $t = 3$ and $t = -5$ . However, since $t$ represents time, a negative value does not make sense in this context. Therefore, we discard $t = -5$ as it is not a physically meaningful solution.

STEP 9

The only physically meaningful solution is $t = 3$ . This is the time when the ball reaches the ground.
The ball reaches the ground after $3$ seconds.

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Find the time $t$ when the ball reaches the ground, given the height function $H = -5(t - 3)(t + 5)$ .

STEP 1

Assumptions
1. The equation $H = -5(t - 3)(t + 5)$ represents the height of a ball over time.
2. The variable $t$ represents time.
3. The ball reaches the ground when the height $H$ is equal to $0$ .

STEP 2

To find when the ball reaches the ground, we need to solve for $t$ when $H = 0$ . We set the equation equal to zero and solve for $t$ .
$0 = -5(t - 3)(t + 5)$

STEP 3

Since the product of the terms is equal to zero, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. We set each factor equal to zero and solve for $t$ .

STEP 4

First, set the first factor equal to zero and solve for $t$ .
$(t - 3) = 0$

STEP 5

Add $3$ to both sides of the equation to solve for $t$ .
$t = 3$

STEP 6

Now, set the second factor equal to zero and solve for $t$ .
$(t + 5) = 0$

STEP 7

Subtract $5$ from both sides of the equation to solve for $t$ .
$t = -5$

STEP 8

We have two solutions for $t$ : $t = 3$ and $t = -5$ . However, since $t$ represents time, a negative value does not make sense in this context. Therefore, we discard $t = -5$ as it is not a physically meaningful solution.

STEP 9

The only physically meaningful solution is $t = 3$ . This is the time when the ball reaches the ground.
The ball reaches the ground after $3$ seconds.

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Find the time ttt when the ball reaches the ground, given the height function H=−5(t−3)(t+5)H = -5(t - 3)(t + 5)H=−5(t−3)(t+5).

STEP 1

Assumptions1. The equation H=−5(t−3)(t+5)H = -5(t - 3)(t + 5)H=−5(t−3)(t+5) represents the height of a ball over time.2. The variable ttt represents time.3. The ball reaches the ground when the height HHH is equal to 000.

STEP 2

To find when the ball reaches the ground, we need to solve for ttt when H=0H = 0H=0. We set the equation equal to zero and solve for ttt.0=−5(t−3)(t+5)0 = -5(t - 3)(t + 5)0=−5(t−3)(t+5)

STEP 3

Since the product of the terms is equal to zero, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. We set each factor equal to zero and solve for ttt.

STEP 4

First, set the first factor equal to zero and solve for ttt.(t−3)=0(t - 3) = 0(t−3)=0

STEP 5

Add 333 to both sides of the equation to solve for ttt.t=3t = 3t=3

STEP 6

Now, set the second factor equal to zero and solve for ttt.(t+5)=0(t + 5) = 0(t+5)=0

STEP 7

Subtract 555 from both sides of the equation to solve for ttt.t=−5t = -5t=−5

STEP 8

We have two solutions for ttt: t=3t = 3t=3 and t=−5t = -5t=−5. However, since ttt represents time, a negative value does not make sense in this context. Therefore, we discard t=−5t = -5t=−5 as it is not a physically meaningful solution.

STEP 9

The only physically meaningful solution is t=3t = 3t=3. This is the time when the ball reaches the ground.The ball reaches the ground after 333 seconds.

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Find the time $t$ when the ball reaches the ground, given the height function $H = -5(t - 3)(t + 5)$ .

Assumptions
1. The equation $H = -5(t - 3)(t + 5)$ represents the height of a ball over time.
2. The variable $t$ represents time.
3. The ball reaches the ground when the height $H$ is equal to $0$ .

To find when the ball reaches the ground, we need to solve for $t$ when $H = 0$ . We set the equation equal to zero and solve for $t$ .
$0 = -5(t - 3)(t + 5)$

Since the product of the terms is equal to zero, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. We set each factor equal to zero and solve for $t$ .

First, set the first factor equal to zero and solve for $t$ .
$(t - 3) = 0$

Add $3$ to both sides of the equation to solve for $t$ .
$t = 3$

Now, set the second factor equal to zero and solve for $t$ .
$(t + 5) = 0$

Subtract $5$ from both sides of the equation to solve for $t$ .
$t = -5$

We have two solutions for $t$ : $t = 3$ and $t = -5$ . However, since $t$ represents time, a negative value does not make sense in this context. Therefore, we discard $t = -5$ as it is not a physically meaningful solution.

The only physically meaningful solution is $t = 3$ . This is the time when the ball reaches the ground.
The ball reaches the ground after $3$ seconds.