A smartphone manufacturer estimates that the total production cost, , in dollars, of assembling phones in a given week can be modeled by a linear function. Data from recent weeks show that producing 400 phones costs \99,000. The factory's weekly budget for production is \$84,000. According to the model, what is the maximum number of phones the factory can assemble in a week without exceeding its production budget?

Solution available with step-by-step explanation

SAT Linear Functions Question 84 | Free SAT Prep | Aniko

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Question 84·Hard·Linear Functions

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A smartphone manufacturer estimates that the total production cost, $C$ , in dollars, of assembling $x$ phones in a given week can be modeled by a linear function. Data from recent weeks show that producing 400 phones costs $54,000 and producing 1,100 phones costs $99,000. The factory's weekly budget for production is $84,000. According to the model, what is the maximum number of phones the factory can assemble in a week without exceeding its production budget?

For linear modeling questions with two data points and a budget or target value, first compute the slope as change in output over change in input, then write a linear equation using point-slope or slope-intercept form. Set the equation equal to (or, if needed, less than or equal to) the budget, solve for the variable, and finally interpret the answer in context—especially checking whether you must round down to keep within a limit or use a whole-number solution.

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Use the two data points

You are told the cost is linear in $x$ and given the cost for 400 phones and 1,100 phones. How can you use these two points to find the cost increase per additional phone?

Write an equation for C in terms of x

Once you know the cost per additional phone (the slope), write a linear equation for $C$ using one of the given points, for example starting from 400 phones and $54,000.

Bring in the budget and think about rounding

Set your cost expression less than or equal to $84,000 and solve for $x$ . When your solution is not a whole number, should you round up or down if the factory must not exceed the budget?

Desmos Guide

Enter the two data points

In Desmos, enter the points $(400,54000)$ and $(1100,99000)$ .

Create the linear model through the points

Define a line through the points by entering y1 = m(x-400)+54000 and then compute m = (99000-54000)/(1100-400) (or just type the fraction directly in place of $m$ ).

Add the budget line

Enter the horizontal line y2 = 84000 to represent the $84,000 budget.

Find the intersection and interpret it

Use the intersection of $y_1$ and $y_2$ to read the $x$ -value (it should be about $866.6$ ). Since $x$ must be a whole number and the cost cannot exceed the budget, choose the greatest integer less than this value: 866.

Step-by-step Explanation

Find the cost per additional phone (slope)

We are told cost is a linear function of the number of phones.

Treat the given data as points on the graph of cost vs. phones: $(400, 54000)$ and $(1100, 99000)$ , where $x$ is phones and $C$ is cost.

The slope $m$ (cost increase per extra phone) is

m = \frac{99000 - 54000}{1100 - 400} = \frac{45000}{700} = \frac{450}{7}.

So each additional phone increases the cost by $\dfrac{450}{7}$ dollars.

Write a linear model for the cost

Use point-slope form with the point $(400, 54000)$ and slope $m = \dfrac{450}{7}$ .

C - 54000 = \frac{450}{7}(x - 400).

This equation gives the production cost $C$ in dollars when $x$ phones are assembled.

Use the budget to set up an equation for x

The factory’s weekly budget is $84,000, and cost increases as more phones are produced, so the maximum number of phones occurs when the model’s cost is equal to the budget.

Set $C = 84000$ in the equation:

84000 - 54000 = \frac{450}{7}(x - 400).

Simplify the left side:

30000 = \frac{450}{7}(x - 400).

Multiply both sides by 7:

210000 = 450(x - 400).

Now divide both sides by 450 and simplify:

x - 400 = \frac{210000}{450} = \frac{1400}{3} \approx 466.6\ldots

x = 400 + \frac{1400}{3} = \frac{2600}{3} \approx 866.6\ldots

Interpret the result to find the maximum whole number of phones

The exact solution to the cost equation is $x \approx 866.6\ldots$ phones. Since you cannot assemble a fraction of a phone and must not exceed the budget, $x$ must be less than or equal to this value.

Therefore, the greatest whole number of phones the factory can assemble without going over the $84,000 budget is 866, which corresponds to choice B.

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Step-by-step Explanation

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Step-by-step Explanation