An engineer wanted to identify the best angle for a cooling fan in an engine in order to get the greatest airflow. The engineer discovered that the function above models the airflow f(\theta), in cubic feet per minute, as a function of the angle of the fan \theta, in degrees. According to the model, what angle, in degrees, gives the greatest airflow? f(\theta) = -0.28(\theta - 27)^2 + 880

C. 27. Choice C is correct. The function $f$ is quadratic, so it will have either a maximum or a minimum at the vertex of the graph. Since the coefficient of the quadratic term ($-0.28$) is negative, the vertex will be at a maximum. The equation $f(\theta) = -0.28(\theta - 27)^2 + 880$ is given in vertex form, so the vertex is at $\theta = 27$. Therefore, an angle of 27 degrees gives the greatest airflow. Choices A and B are incorrect and may be the result of misidentifying which value in a quadratic equation in vertex form represents the vertex. Choice D is incorrect. This choice identifies the maximum value of $f(\theta)$ rather than the value of $\theta$ for which $f(\theta)$ is maximized.

SAT Math skill page

SAT Quadratic Word Problems Practice

Translate word problems into quadratics, then use the form that answers the question fastest.

Practice quadratic word problems in Dolphin SAT Practice questions

12-18 min practice time3 questions on pageAdvanced Math

Practice time12-18 min

Question bank3 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Build quadratic equations from word problems
Interpret zeros, vertex, and intercepts in context
Choose factored, standard, or vertex form
Check whether the answer makes sense in the situation

Dolphin question bank

Practice your skill

Question 1 of 3Hard

A machine launches a softball from ground level. The softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds and hits the ground at 3.6 seconds. Which equation represents the height above ground h, in meters, of the softball t seconds after it is launched?

$h = - t^{2} + 3.6$
$h = - t^{2} + 51.84$
$h = - 16 (t + 51.84)^{2} - 3.6$
$h = - 16 (t - 1.8)^{2} + 51.84$

Show answer and solution

Correct answer $D . h = - 16 (t - 1.8)^{2} + 51.84$

Solution walkthrough

An equation representing the height above ground $h$ , in meters, of a softball $t$ seconds after it is launched by a machine from ground level can be written in the form $h = - a (t - b)^{2} + c$ , where $a$ , $b$ , and $c$ are positive constants. In this equation, $b$ represents the time, in seconds, at which the softball reaches its maximum height of $c$ meters above the ground.
It's given that this softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds; therefore, $b = 1.8$ and $c = 51.84$ . Substituting 1.8 for $b$ and 51.84 for $c$ in the equation $h = - a (t - b)^{2} + c$ yields $h = - a (t - 1.8)^{2} + 51.84$ .
It's also given that this softball hits the ground at 3.6 seconds; therefore, $h = 0$ when $t = 3.6$ . Substituting 0 for $h$ and 3.6 for $t$ in the equation $h = - a (t - 1.8)^{2} + 51.84$ yields $0 = - a (3.6 - 1.8)^{2} + 51.84$ , which is equivalent to $0 = - a (1.8)^{2} + 51.84$ , or $0 = - 3.24 a + 51.84$ .
Adding 3.24a to both sides of this equation yields $3.24 a = 51.84$ . Dividing both sides of this equation by 3.24 yields $a = 16$ .
Substituting 16 for $a$ in the equation $h = - a (t - 1.8)^{2} + 51.84$ yields $h = - 16 (t - 1.8)^{2} + 51.84$ . Therefore, $h = - 16 (t - 1.8)^{2} + 51.84$ represents the height above ground $h$ , in meters, of this softball $t$ seconds after it is launched.

Why the other choices miss

Choice A is incorrect. This equation represents a situation where the maximum height is 3.6 meters above the ground at 0 seconds, not 51.84 meters above the ground at 1.8 seconds. Choice B is incorrect. This equation represents a situation where the maximum height is 51.84 meters above the ground at 0 seconds, not 1.8 seconds. Choice C is incorrect and may result from conceptual or calculation errors.

Question 2 of 3Medium

An engineer wanted to identify the best angle for a cooling fan in an engine in order to get the greatest airflow. The engineer discovered that the function above models the airflow f(

θ)

, in cubic feet per minute, as a function of the angle of

t h e f an θ

, in degrees. According to the model, what angle, in degrees, gives the greatest airflow?

f (θ) = - 0.28 (θ - 27)^{2} + 880

-0.28
0.28
27
880

Show answer and solution

Correct answerC. 27

Solution walkthrough

The function $f$ is quadratic, so it will have either a maximum or a minimum at the vertex of the graph.
Since the coefficient of the quadratic term ( $- 0.28$ ) is negative, the vertex will be at a maximum.
The equation $f (θ) = - 0.28 (θ - 27)^{2} + 880$ is given in vertex form, so the vertex is at $θ = 27$ .
Therefore, an angle of 27 degrees gives the greatest airflow.

Why the other choices miss

Choices A and B are incorrect and may be the result of misidentifying which value in a quadratic equation in vertex form represents the vertex. Choice D is incorrect. This choice identifies the maximum value of

f (θ)

rather than the value of

θ

for which

f (θ)

is maximized.

Question 3 of 3Hard

A rectangle has an area of 155 square inches. The length of the rectangle is 4 inches less than 7 times the width of the rectangle. What is the width of the rectangle, in inches?

Show answer and solution

Correct answerC. 5

Solution walkthrough

Let $x$ represent the width, in inches, of the rectangle. It's given that the length of the rectangle is 4 inches less than 7 times its width, or $7 x - 4$ inches.
The area of a rectangle is equal to its width multiplied by its length. Multiplying the width, $x$ inches, by the length, $7 x - 4$ inches, yields $x (7 x - 4)$ square inches.
It's given that the rectangle has an area of 155 square inches, so it follows that $x (7 x - 4) = 155$ , or $7 x^{2} - 4 x = 155$ . Subtracting 155 from both sides of this equation yields $7 x^{2} - 4 x - 155 = 0$ .
Factoring the left-hand side of this equation yields $(7 x + 31) (x - 5) = 0$ . Applying the zero product property to this equation yields two solutions: $x = - \frac{31}{7}$ and $x = 5$ .
Since $x$ is the rectangle’s width, in inches, which must be positive, the value of $x$ is 5. Therefore, the width of the rectangle, in inches, is 5.

Avoid these traps

Common mistakes on this skill

Keeping every algebraic root without checking context

Time, length, and quantity restrictions can make one mathematical root impossible in the situation.

Confusing a zero with the vertex

Zeros mark where the modeled output is 0; the vertex marks a maximum or minimum.

Study plan

How to practice this skill in Dolphin

Define the variable and identify what the output represents.
Choose a quadratic form that exposes the needed feature: zeros, vertex, or initial value.
Solve and then reject any result that violates the units, domain, or physical context.

Practice quadratic word problems in Dolphin SAT

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FAQ

Questions about SAT Quadratic Word Problems Practice

Which quadratic form is best for a word problem?

Factored form exposes zeros, vertex form exposes a maximum or minimum, and standard form exposes the initial value when x = 0.

Why might a quadratic model produce an invalid root?

The equation extends beyond the real situation, where negative time, length, or count may be impossible.