The value , in thousands of dollars, of an antique car years after 2000 is modeled by According to the model, for which integer values of will the car be worth at least \$28,000?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 122 | Free SAT Prep | Aniko

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Question 122·Medium·Nonlinear Functions

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The value $V$ , in thousands of dollars, of an antique car $t$ years after 2000 is modeled by

V = -0.25(t - 10)^2 + 30.

According to the model, for which integer values of $t$ will the car be worth at least $28,000?

For quadratic inequalities written in vertex form, first translate the word condition (such as at least a certain amount) into an inequality using the function expression. Isolate the squared term, paying close attention to signs and flipping the inequality if you divide by a negative. Then take square roots to find a numerical interval for the variable, approximate roots if needed, and finally restrict to integers or any other domain the question specifies before matching the result to the answer choices.

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Set up the inequality for the value

The car’s value is given by $V = -0.25(t - 10)^2 + 30$ , and $V$ is in thousands of dollars. How do you write “at least 28 thousand dollars” as an inequality involving $V$ ?

Substitute and rearrange

After you write $V \ge 28$ , substitute the expression for $V$ and then move constants so that the squared term $(t - 10)^2$ is isolated on one side.

Be careful with the negative coefficient

When you divide or multiply both sides of an inequality by a negative number (like $-0.25$ ), what happens to the inequality sign? Apply that rule before taking square roots.

From real solutions to integer solutions

Once you find the interval of $t$ values that satisfy the inequality, remember that the question asks for integer values only. Which integers fall inside that interval?

Desmos Guide

Graph the value function

In Desmos, enter the equation y = -0.25(x - 10)^2 + 30 to represent the car’s value (in thousands) as a function of $t$ (use $x$ for $t$ ).

Graph the comparison value

Enter the horizontal line y = 28 to represent a value of 28 thousand dollars.

Find where the value is at least 28

Look at the intersection points of the parabola and the line $y = 28$ . Identify the $x$ -coordinates of these two intersections, then determine all integer $x$ -values between them (inclusive); these are the integer $t$ values where the car’s value is at least 28 thousand dollars.

Step-by-step Explanation

Translate the wording into an inequality

The value $V$ is given in thousands of dollars.

At least $28{,}000 means the value is greater than or equal to 28 (in thousands):

V \ge 28.

Since $V = -0.25(t - 10)^2 + 30$ , substitute this into the inequality:

-0.25(t - 10)^2 + 30 \ge 28.

Isolate the squared expression

Solve the inequality step by step:

Subtract 30 from both sides:

-0.25(t - 10)^2 + 30 - 30 \ge 28 - 30

-0.25(t - 10)^2 \ge -2.

Now divide both sides by $-0.25$ . Remember, dividing by a negative number reverses the inequality sign:

(t - 10)^2 \le 8.

Solve the inequality $(t - 10)^2 \le 8$

Take square roots of both sides. When you take the square root of a squared expression, you get a plus-minus:

|t - 10| \le \sqrt{8}.

This means $t$ is within a distance of $\sqrt{8}$ from 10:

10 - \sqrt{8} \le t \le 10 + \sqrt{8}.

Approximate $\sqrt{8}$ :

\sqrt{8} \approx 2.83.

So the bounds for $t$ are approximately:

10 - 2.83 \le t \le 10 + 2.83

7.17 \le t \le 12.83.

Select the integer values of $t$ and match the choice

The model is defined for real $t$ , but the question asks for integer values of $t$ .

From $7.17 \le t \le 12.83$ , the integers that satisfy this are

8, 9, 10, 11, 12.

In inequality form, this is

8 \le t \le 12,

which corresponds to answer choice D.

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Step-by-step Explanation

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Step-by-step Explanation