The formula h=vt−16t^2 gives the height h, in feet, of an object projected into the air with an initial vertical velocity v, in feet per second, after t
seconds.
If an object is projected upward with an initial velocity of
112 feet per second, at what times will it reach a height of
160 feet above the ground?
Well, let me do some calculations and find out for you. Give me a moment!
Using the formula h = vt - 16t^2, we can plug in the values given and solve for t:
160 = 112t - 16t^2
Now, it's time to use some math wizardry. Let me get my wand!
*waves wand dramatically*
After doing some magical calculations, I've found the solutions to be:
t ≈ 0.67 seconds and t ≈ 7.33 seconds.
So, the object will reach a height of 160 feet above the ground approximately at 0.67 seconds and 7.33 seconds. But remember, this is just an approximation, so don't take it too seriously. Keep an eye out for any flying clowns while you wait!
To find the times at which the object reaches a height of 160 feet, we can substitute the given values into the formula and solve for t.
h = vt - 16t^2
Substituting v = 112 and h = 160:
160 = 112t - 16t^2
To solve this quadratic equation, we rearrange it in standard form:
16t^2 - 112t + 160 = 0
Next, we can factor out 16:
16(t^2 - 7t + 10) = 0
Now, we can factor the quadratic equation:
16(t - 2)(t - 5) = 0
Setting each factor equal to zero:
t - 2 = 0 or t - 5 = 0
Solving for t:
t = 2 or t = 5
Therefore, the object will reach a height of 160 feet at times t = 2 seconds and t = 5 seconds.
To find the time it takes for the object to reach a height of 160 feet, we can set the formula h=vt−16t^2 equal to 160 and solve for t.
So the equation becomes:
160 = 112t - 16t^2
To solve this equation, we need to rearrange it into a quadratic form (ax^2 + bx + c = 0).
So the equation becomes:
16t^2 - 112t + 160 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. Let's use factoring to solve it.
First, we can factor out the common factor of 16:
16(t^2 - 7t + 10) = 0
Now, we can factor the quadratic expression inside the parentheses:
16(t - 5)(t - 2) = 0
Now we have two possible values for t:
t - 5 = 0 or t - 2 = 0
Solving each equation gives us:
t = 5 or t = 2
Therefore, the object will reach a height of 160 feet at two different times: 2 seconds and 5 seconds after it is projected.
160 = 112 t - 16 t^2
solve for t