h=235t-16t^2
find all values of t for which the rocket's height is 151 feet
151 = 235t - 16t^2
16t^2 - 235t + 151 = 0
Solve using the methods you learned to solve quadratics
There will be two positive answers, one would represent the time on its way down, and the other on its path down
(According to your equation, the rocket has an initial velocity of 235 ft/s, but has no propulsion other than gravity after launch.)
To find all values of t for which the rocket's height is 151 feet, we'll set the equation h = 151 and solve for t.
The equation h = 235t - 16t^2 represents the height of the rocket at a given time t. We substitute h with 151 and solve the resulting quadratic equation.
151 = 235t - 16t^2
To solve this equation, we'll rearrange it to standard quadratic form:
16t^2 - 235t + 151 = 0
Now we have a quadratic equation of the form ax^2 + bx + c = 0, where:
a = 16
b = -235
c = 151
There are a few methods to solve this quadratic equation, such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying this formula to our equation, we have:
t = (-(-235) ± √((-235)^2 - 4 * 16 * 151)) / (2 * 16)
Simplifying this expression:
t = (235 ± √(55225 - 9664)) / 32
t = (235 ± √(45561)) / 32
The discriminant inside the square root (√(45561)) is equal to 213, which means we have two distinct real solutions for t.
Applying the square root to the discriminant:
t = (235 ± 213) / 32
This gives us two possible values for t:
t₁ = (235 + 213) / 32 = 448/32 = 14
t₂ = (235 - 213) / 32 = 22/32 = 11/16
Therefore, the rocket's height is 151 feet at two different times: t = 14 and t = 11/16.