A nonprofit is ordering custom candles for a fundraiser. The supplier charges \3 for each additional candle. There is also a one-time shipping fee of \cnn \ge 120$?

Solution available with step-by-step explanation

SAT Linear Functions Question 138 | Free SAT Prep | Aniko

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Question 138·Hard·Linear Functions

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A nonprofit is ordering custom candles for a fundraiser. The supplier charges $4 per candle for the first 120 candles and $3 for each additional candle. There is also a one-time shipping fee of $90 for the entire order. Which function $c$ gives the total cost, in dollars, of ordering $n$ candles, where $n \ge 120$ ?

For tiered-pricing problems, separate the cost into a fixed part and a variable part. First compute the total cost at the breakpoint (here, 120 candles) including any one-time fees. Then add the cost for the extra items using (extra quantity) $\times$ (marginal rate), i.e., $3(n-120)$ . Finally, simplify to get a linear function in $n$ and match it to the choices.

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Start with the cost at 120 candles

Compute the total cost of exactly 120 candles, including the $90 shipping fee.

Represent the extra candles

For $n\ge120$ , how many candles are beyond 120? Express that number using $n$ .

Build the expression

Total cost $=$ (cost for first 120 and shipping) $+$ $3 times the number of extra candles. Then simplify.

Desmos Guide

Compute the point at $n=120$

In Desmos, calculate the total cost for 120 candles: enter 4*120+90 to get the cost value (this is $c(120)$ ).

Use the slope for $n\ge120$

Since each additional candle beyond 120 costs $3, the graph for $n\ge120$ must have slope $3$ . Think of the line passing through the point $(120, c(120))$ with slope $3$ .

Check which option matches that line

Graph each answer choice as a function (e.g., c1(n)=3n+570, c2(n)=4n+90, etc.). Then evaluate each at $n=120$ (use a table or type c1(120), c2(120), etc.) and confirm which one also has slope $3$ . The one that matches the correct value at $120$ and has slope $3$ is $c(n)=3n+210$ .

Step-by-step Explanation

Compute the fixed cost at 120 candles

For the first 120 candles, the supplier charges $4 each, and there is a one-time shipping fee of $90.

Fixed cost at 120 candles:

4\cdot120+90

Add the cost of additional candles

If $n\ge120$ , then there are $n-120$ additional candles beyond 120, each costing $3.

Additional cost:

3(n-120)

So the total cost is:

4\cdot120+90+3(n-120)

Simplify and match to a choice

Simplify:

\begin{aligned} 4\cdot120+90+3(n-120) &=480+90+3n-360\\ &=3n+210 \end{aligned}

Therefore, the function is $c(n)=3n+210$ .

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Step-by-step Explanation

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