A ball is launched straight upward. Its height relative to the ground, in meters (negative values indicate below ground), is modeled by the function where is the time, in seconds, after the ball is launched. How many seconds after the ball is launched does it first reach the ground?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 123 | Free SAT Prep | Aniko

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Question 123·Hard·Nonlinear Functions

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A ball is launched straight upward. Its height relative to the ground, in meters (negative values indicate below ground), is modeled by the function

h(t)= -\tfrac{1}{20}\,(t-18)\bigl(t^2+4t-45\bigr),

where $t$ is the time, in seconds, after the ball is launched.

How many seconds after the ball is launched does it first reach the ground?

Your answer

For nonlinear motion problems like this, immediately translate the physical question into an equation, usually by setting the height function equal to the desired height (often 0 for ground). Factor the function if possible, use the zero-product property to find all algebraic solutions, and then filter them using the real-world context (time must be nonnegative, distances nonnegative, etc.). Finally, if the question asks for "first" or "earliest," choose the smallest valid time; this is faster and less error-prone than expanding everything or using the quadratic formula unnecessarily.

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Connect ground level to the function

What value of $h(t)$ corresponds to the ball being at ground level? Set the given function equal to that value.

Use the structure of the function

You are given $h(t)$ as a product of factors. How can you use the fact that a product equals 0 to find possible values of $t$ ?

Consider the context of time

After you find all values of $t$ that make $h(t)=0$ , think about which of those times are allowed by the phrase "seconds after the ball is launched" and which comes first.

Desmos Guide

Enter the height function

Type h(t) = -1/20*(t-18)*(t^2+4t-45) into Desmos. If Desmos prefers x instead of t, replace every t with x.

View the graph and identify intersections with the ground

Graph $y=h(t)$ and look for the points where the graph crosses the horizontal axis $y=0$ (these are the times when the ball is at ground level).

Find the earliest nonnegative intersection

Use the cursor or tap on each intersection point to see its $t$ - (or $x$ -) coordinate. Among the intersection points with $t\ge 0$ , identify the smallest value; that value is the time when the ball first reaches the ground.

Step-by-step Explanation

Translate the question into an equation

The ball is at ground level when its height is 0. So we need to solve the equation

-\tfrac{1}{20}(t-18)(t^2+4t-45)=0

for $t$ , and then decide which solution makes sense in the context (time after launch).

Use the zero-product property

A product is zero if and only if at least one factor is zero. The factor $-\tfrac{1}{20}$ is never zero, so we only need to look at

$t-18=0$ , or
$t^2+4t-45=0$ .

From $t-18=0$ , we immediately get one solution: $t=18$ .

Next, we must solve $t^2+4t-45=0$ . This quadratic is factorable.

Factor the quadratic and find all roots

To factor $t^2+4t-45$ , we look for two numbers that multiply to $-45$ and add to $4$ . Those numbers are $9$ and $-5$ , so

t^2+4t-45 = (t+9)(t-5).

Now the full factored form of $h(t)$ is

h(t) = -\tfrac{1}{20}(t-18)(t+9)(t-5).

Setting $h(t)=0$ gives three equations:

$t-18=0$ → $t=18$
$t+9=0$ → $t=-9$
$t-5=0$ → $t=5$ .

So algebraically, the three times when the height is 0 are $t=-9$ , $t=5$ , and $t=18$ .

Choose the time that matches the situation

The variable $t$ represents time after the ball is launched, so only $t\ge 0$ makes sense.

$t=-9$ is negative, so it is not physically meaningful in this context.
$t=5$ and $t=18$ are both nonnegative, so they are possible times when the ball is at ground level.

The question asks when the ball first reaches the ground after launch, so we take the smaller of these two valid times.

Therefore, the ball first reaches the ground $5$ seconds after it is launched.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation