Question

H=204t-16t^2
Find all the values of t for which the rocket’s height is 100ft

Answer 1

The values of
`\(t\)` when the rocket's height is 100 feet, calculated using the quadratic formula, are approximately
`\(12.24\)` seconds or
`\(0.51\)` seconds (rounded to the nearest hundredth).

To find the values of
`\( t \)` when the rocket's height is 100 feet, we'll use the equation given for the rocket's height:

`\[ h = 204t - 16t^2 \]`

Given that the height
`(\( h \))` is 100 feet, set up the equation:

`\[ 100 = 204t - 16t^2 \]`

Rewrite the equation in standard quadratic form:

`\[ 16t^2 - 204t + 100 = 0 \]`

Now, to solve for
`\( t \)` we can use the quadratic formula:

`\[ t = (-b \pm √(b^2 - 4ac))/(2a) \]`

For the quadratic equation
`\(16t^2 - 204t + 100 = 0\),` the values of
`\(a\)`,
`\(b\)`, and
`\(c\)`are:

`\[ a = 16, \quad b = -204, \quad c = 100 \]`

Now, let's substitute these values into the quadratic formula:

`\[ t = (-(-204) \pm √((-204)^2 - 4 \cdot 16 \cdot 100))/(2 \cdot 16) \]`

Solving this equation will provide the values of
`\( t \)` when the rocket's height is 100 feet. Let's calculate it.

Apologies for the confusion earlier. Let's solve the quadratic equation
`\(16t^2 - 204t + 100 = 0\)` to find the values of \(t\) when the rocket's height is 100 feet.

Using the quadratic formula:

`\[ t = (-b \pm √(b^2 - 4ac))/(2a) \]`

Given
`\(a = 16\)`,
`\(b = -204\)`, and
`\(c = 100\)` , let's plug these values into the formula:

`\[ t = (-(-204) \pm √((-204)^2 - 4 \cdot 16 \cdot 100))/(2 \cdot 16) \]`

Calculating the values:

`\[ t = (204 \pm √(41616 - 6400))/(32) \]`

`\[ t = (204 \pm √(35216))/(32) \]`

`\[ t = (204 \pm 187.75)/(32) \]`

This results in two solutions for
`\(t\)`:

`\[ t_1 = (204 + 187.75)/(32) = (391.75)/(32) \approx 12.24 \]`

`\[ t_2 = (204 - 187.75)/(32) = (16.25)/(32) \approx 0.51 \]`

Therefore, when the rocket's height is 100 feet, the values of
`\(t\)` are approximately
`\(12.24\)` seconds or
`\(0.51\)` seconds (rounded to the nearest hundredth).

complete the question

A model rocket is launched with an initial upward velocity of 204 ft/s. The rocket's height h in feet after t seconds is given by the following equation: h = 204t - 16t^2. Find all values of t for which the rocket's height is 100 feet. Round your answer(s) to the nearest hundredth. (If there is more than one answer, use the "or" button.)