Before you read

Answer first — see what you already know.

A rocket is launched upward from a platform with height modeled by $h (t) = - 16 t^{2} + 64 t + 80$ feet. What is the maximum height the rocket reaches?

A
$80$ feet
B
$144$ feet
C
$128$ feet
D
$64$ feet

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A tennis ball is hit from ground level with $h (t) = - 16 t^{2} + 48 t$ . After how many seconds does it return to the ground?

A
$1.5$ seconds
B
$2$ seconds
C
$3$ seconds
D
$4$ seconds

Check your answer

Answer:

Theory

Common Quadratic Models on the SAT

Quadratics model many real-world situations:

1. Projectile Motion (the most common!):
$h(t) = -16t^2 + v_0 t + h_0$
where $h$ = height (feet), $t$ = time (seconds), $v_0$ = initial velocity, $h_0$ = initial height.
(In metric: $-4.9t^2$ instead of $-16t^2$ .)

Projectile path: maximum height at vertex

2. Area Problems: length $\times$ width, often with a constraint.

3. Revenue/Profit: $R = (\text{price})(\text{quantity})$ , where one depends on the other.

4. Number Problems: "two numbers whose product/sum satisfies a condition."

Theory

Projectile Motion

For $h(t) = -16t^2 + v_0 t + h_0$ :

Question	How to find it
Initial height	$h(0) = h_0$
When does it hit the ground?	Solve $h(t) = 0$
Maximum height	$t_{\text{max}} = \frac{v_0}{32}$ , then compute $h(t_{\text{max}})$
When does it reach height $k$ ?	Solve $h(t) = k$

Example 1

A ball is thrown upward from a 48-foot building with an initial velocity of 32 ft/s. $h(t) = -16t^2 + 32t + 48$ . When does it hit the ground?

Set $h(t) = 0$ : $-16t^2 + 32t + 48 = 0$

Divide by $-16$ : $t^2 - 2t - 3 = 0$

Factor: $(t - 3)(t + 1) = 0$

$t = 3$ or $t = -1$ (reject negative time)

The ball hits the ground at $t = 3$ seconds.

t = 3

seconds

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2 more sections including examples, practice problems, and step-by-step solutions.

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