Question

A model rocket is launched with an initial upward velocity of 164 ft/s. The rocket's height (in feet) after t seconds is given by the following equation: \( h = 164t - 16t^2 \). Find all values of t for which the rocket's height is 92 feet.

A) \( t = 1 \)
B) \( t = 2 \)
C) \( t = 3 \)
D) \( t = 4 \)

Answer 1

The values of t for which the rocket's height is 92 feet are approximately:


`A) \( t \approx 1.18 \)\\B) \( t \approx 5.07 \)`

To find the values of t for which the rocket's height is 92 feet, we need to solve the equation
`\( h = 164t - 16t^2 \)` when h is equal to 92.

So, we can write the equation as
`\( 92 = 164t - 16t^2 \).`

To solve this quadratic equation, we can rearrange it to
`\( 16t^2 - 164t + 92 = 0 \).`

Now, we can factorize or use the quadratic formula to find the values of t.

Using the quadratic formula, which states that for an equation of the form
`\( ax^2 + bx + c = 0 \)`, the solutions for x are given by
`\( x = (-b \pm √(b^2 - 4ac))/(2a) \)`, we can find the values of t.

For our equation, a = 16, b = -164, and c = 92.

Substituting these values into the quadratic formula, we have:


`\( t = (-(-164) \pm √((-164)^2 - 4(16)(92)))/(2(16)) \).`

Simplifying the expression inside the square root, we get:


`\( t = (164 \pm √(26896 - 5888))/(32) \).`

Further simplifying, we have:


`\( t = (164 \pm √(21008))/(32) \).`

Now, we can simplify the square root:


`\( t = (164 \pm √(4 * 5252))/(32) \).\( t = (164 \pm 2√(5252))/(32) \).\( t = (164 \pm 2√(4 * 1313))/(32) \).\( t = (164 \pm 2√(4) √(1313))/(32) \).\( t = (164 \pm 4 √(1313))/(32) \).\\`
Simplifying further, we have:


`\( t = (41 \pm √(1313))/(8) \).`