Solved on Dec 05, 2023

Find all times (in seconds) when a ball thrown with initial height 6 ft and velocity 44 ft/s reaches a height of 26 ft. Solve the quadratic equation $h = 6 + 44t - 16t^2 = 26$ to find $t$ .

STEP 1

Assumptions
1. The initial height from which the ball is thrown is 6 feet.
2. The initial upward velocity of the ball is $44 \mathrm{ft/s}$ .
3. The acceleration due to gravity is $-32 \mathrm{ft/s^2}$ (which is doubled to $-16 \mathrm{ft/s^2}$ in the equation to match the standard form of a quadratic equation).
4. The height $h$ of the ball after $t$ seconds is given by the quadratic equation $h = 6 + 44t - 16t^2$ .
5. We are looking for the time $t$ when the height $h$ is 26 feet.

STEP 2

Set the height $h$ to 26 feet in the quadratic equation to find the values of $t$ .
$26 = 6 + 44t - 16t^2$

STEP 3

Rearrange the equation to set it to zero, which is the standard form for solving quadratic equations.
$16t^2 - 44t + (6 - 26) = 0$

STEP 4

Simplify the constant term.
$16t^2 - 44t - 20 = 0$

STEP 5

Divide the entire equation by 4 to simplify the coefficients, making it easier to solve.
$4t^2 - 11t - 5 = 0$

STEP 6

Now we need to solve the quadratic equation. Since the roots are irrational, we will use the quadratic formula:
$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 4$ , $b = -11$ , and $c = -5$ .

STEP 7

Calculate the discriminant, which is the part under the square root in the quadratic formula:
$\Delta = b^2 - 4ac$

STEP 8

Plug in the values for $a$ , $b$ , and $c$ to calculate the discriminant.
$\Delta = (-11)^2 - 4(4)(-5)$

STEP 9

Simplify the discriminant.
$\Delta = 121 + 80$

STEP 10

Calculate the discriminant.
$\Delta = 201$

STEP 11

Now that we have the discriminant, we can find the two possible values for $t$ using the quadratic formula.
$t = \frac{-(-11) \pm \sqrt{201}}{2(4)}$

STEP 12

Simplify the numerator of the quadratic formula.
$t = \frac{11 \pm \sqrt{201}}{8}$

STEP 13

Calculate the two possible values for $t$ by considering both the positive and negative square roots.
For the positive root:
$t = \frac{11 + \sqrt{201}}{8}$

STEP 14

Calculate the value of $t$ using the positive root.
$t \approx \frac{11 + 14.177}{8}$

STEP 15

Simplify the expression.
$t \approx \frac{25.177}{8}$

STEP 16

Calculate the approximate value of $t$ with the positive root.
$t \approx 3.147$

STEP 17

Now calculate the value of $t$ using the negative root.
$t = \frac{11 - \sqrt{201}}{8}$

STEP 18

Calculate the value of $t$ using the negative root.
$t \approx \frac{11 - 14.177}{8}$

STEP 19

Simplify the expression.
$t \approx \frac{-3.177}{8}$

STEP 20

Calculate the approximate value of $t$ with the negative root.
$t \approx -0.397$

STEP 21

Since time cannot be negative in this context, we discard the negative value of $t$ .

STEP 22

Round the positive value of $t$ to the nearest hundredth.
$t \approx 3.15 \text{ seconds}$
The ball's height is 26 feet at approximately $t = 3.15$ seconds.

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Find all times (in seconds) when a ball thrown with initial height 6 ft and velocity 44 ft/s reaches a height of 26 ft. Solve the quadratic equation h=6+44t−16t2=26h = 6 + 44t - 16t^2 = 26h=6+44t−16t2=26 to find ttt.

STEP 1

STEP 2

Set the height hhh to 26 feet in the quadratic equation to find the values of ttt.26=6+44t−16t226 = 6 + 44t - 16t^226=6+44t−16t2

STEP 3

Rearrange the equation to set it to zero, which is the standard form for solving quadratic equations.16t2−44t+(6−26)=016t^2 - 44t + (6 - 26) = 016t2−44t+(6−26)=0

STEP 4

Simplify the constant term.16t2−44t−20=016t^2 - 44t - 20 = 016t2−44t−20=0

STEP 5

Divide the entire equation by 4 to simplify the coefficients, making it easier to solve.4t2−11t−5=04t^2 - 11t - 5 = 04t2−11t−5=0

STEP 6

Now we need to solve the quadratic equation. Since the roots are irrational, we will use the quadratic formula:t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}t=2a−b±b2−4ac​​where a=4a = 4a=4, b=−11b = -11b=−11, and c=−5c = -5c=−5.

STEP 7

Calculate the discriminant, which is the part under the square root in the quadratic formula:Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac

STEP 8

Plug in the values for aaa, bbb, and ccc to calculate the discriminant.Δ=(−11)2−4(4)(−5)\Delta = (-11)^2 - 4(4)(-5)Δ=(−11)2−4(4)(−5)

STEP 9

Simplify the discriminant.Δ=121+80\Delta = 121 + 80Δ=121+80

STEP 10

Calculate the discriminant.Δ=201\Delta = 201Δ=201

STEP 11

Now that we have the discriminant, we can find the two possible values for ttt using the quadratic formula.t=−(−11)±2012(4)t = \frac{-(-11) \pm \sqrt{201}}{2(4)}t=2(4)−(−11)±201​​

STEP 12

Simplify the numerator of the quadratic formula.t=11±2018t = \frac{11 \pm \sqrt{201}}{8}t=811±201​​

STEP 13

Calculate the two possible values for ttt by considering both the positive and negative square roots.For the positive root:t=11+2018t = \frac{11 + \sqrt{201}}{8}t=811+201​​

STEP 14

Calculate the value of ttt using the positive root.t≈11+14.1778t \approx \frac{11 + 14.177}{8}t≈811+14.177​

STEP 15

Simplify the expression.t≈25.1778t \approx \frac{25.177}{8}t≈825.177​

STEP 16

Calculate the approximate value of ttt with the positive root.t≈3.147t \approx 3.147t≈3.147

STEP 17

Now calculate the value of ttt using the negative root.t=11−2018t = \frac{11 - \sqrt{201}}{8}t=811−201​​

STEP 18

Calculate the value of ttt using the negative root.t≈11−14.1778t \approx \frac{11 - 14.177}{8}t≈811−14.177​

STEP 19

Simplify the expression.t≈−3.1778t \approx \frac{-3.177}{8}t≈8−3.177​

STEP 20

Calculate the approximate value of ttt with the negative root.t≈−0.397t \approx -0.397t≈−0.397

STEP 21

Since time cannot be negative in this context, we discard the negative value of ttt.

STEP 22

Round the positive value of ttt to the nearest hundredth.t≈3.15 secondst \approx 3.15 \text{ seconds}t≈3.15 secondsThe ball's height is 26 feet at approximately t=3.15t = 3.15t=3.15 seconds.

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Find all times (in seconds) when a ball thrown with initial height 6 ft and velocity 44 ft/s reaches a height of 26 ft. Solve the quadratic equation $h = 6 + 44t - 16t^2 = 26$ to find $t$ .

Set the height $h$ to 26 feet in the quadratic equation to find the values of $t$ .
$26 = 6 + 44t - 16t^2$

Rearrange the equation to set it to zero, which is the standard form for solving quadratic equations.
$16t^2 - 44t + (6 - 26) = 0$

Simplify the constant term.
$16t^2 - 44t - 20 = 0$

Divide the entire equation by 4 to simplify the coefficients, making it easier to solve.
$4t^2 - 11t - 5 = 0$

Now we need to solve the quadratic equation. Since the roots are irrational, we will use the quadratic formula:
$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 4$ , $b = -11$ , and $c = -5$ .

Calculate the discriminant, which is the part under the square root in the quadratic formula:
$\Delta = b^2 - 4ac$

Plug in the values for $a$ , $b$ , and $c$ to calculate the discriminant.
$\Delta = (-11)^2 - 4(4)(-5)$

Simplify the discriminant.
$\Delta = 121 + 80$

Calculate the discriminant.
$\Delta = 201$

Now that we have the discriminant, we can find the two possible values for $t$ using the quadratic formula.
$t = \frac{-(-11) \pm \sqrt{201}}{2(4)}$

Simplify the numerator of the quadratic formula.
$t = \frac{11 \pm \sqrt{201}}{8}$

Calculate the two possible values for $t$ by considering both the positive and negative square roots.
For the positive root:
$t = \frac{11 + \sqrt{201}}{8}$

Calculate the value of $t$ using the positive root.
$t \approx \frac{11 + 14.177}{8}$

Simplify the expression.
$t \approx \frac{25.177}{8}$

Calculate the approximate value of $t$ with the positive root.
$t \approx 3.147$

Now calculate the value of $t$ using the negative root.
$t = \frac{11 - \sqrt{201}}{8}$

Calculate the value of $t$ using the negative root.
$t \approx \frac{11 - 14.177}{8}$

Simplify the expression.
$t \approx \frac{-3.177}{8}$

Calculate the approximate value of $t$ with the negative root.
$t \approx -0.397$

Since time cannot be negative in this context, we discard the negative value of $t$ .

Round the positive value of $t$ to the nearest hundredth.
$t \approx 3.15 \text{ seconds}$
The ball's height is 26 feet at approximately $t = 3.15$ seconds.