The classic "when does x enter the y" type question

Question

I need help with a certain question in a math puzzle. The equation is h(t)=-16t^2+24t+48. I need to find out when it'll reach the max height.

Answer 1

Hi

At t = 3/4

Given h(t) = - 16t² + 24t + 48

1. Notice this fits in the standard form of a parabola

f(x) = ax² + bx + c

2. You have value for the coefficients of x² and x that you can use to find the vertex of your parabola

3.There is certain information associated with the sign for the coefficient of x² that defines which direction your parabola opens

4. The coefficients of x² and x can be used to determine the x coordinate of the vertex

5. Once the x coordinate of the vertex is found, plug it back into your function to determine y coordinate of the vertex.

h(t) = - 16t² + 24t + 48

Lets list the coefficients relative to the standard format f(x) = ax² + bx + c

a = -16, b = 24 and c = 48

a is negative, so your parabola opens downward and the vertex associated with it is the maximum

Coordinates for the vertex for a parabola in standard form are

x coordinate = -b/2a = -24/2(-16) = -24/-32 = 24/32 = 3/4

Plug in the x coordinate to determine y or h(t)

y = h(3/4) = -16(3/4)² + 24(3/4) + 48

y = h(3/4) = -16(9/16) + 18 + 48

y = h(3/4) = -9 + 18 + 48

y = h(3/4) = 57

So the coordinates of the maxima or vertex for h(t) = -16t² + 24t + 48

x = 3/4

y = 57

(3/4, 57)

The x coordinate represents the time, t

You can graph your function at Desmos.com to confirm the coordinates of the vertex.

Answer 2

You can determine the answer to this by studying the quadratic formula. The formula states that the roots (aka where the function touches the x-axis) can be determined by -(b+/- sqrt(b² - 4ac))/2a.

If we break up this formula a bit, you can see that one part is the -b/2a and the other part is the +/-sqrt(b² - 4ac))/2a. This suggests then that the center of any parabola must be at -b/2a when the function is written as ax² + bx + c. In this problem the -b/2a = -24/(2)*(-16) or 3/4. So the peak must occur when the x = 0.75. To determine the y-value, plug 0.75 into the function and voila!!