At a charity festival, the first 50 admission tickets sold raise \18 each. Let represent the total number of tickets sold, where . Which function gives the total amount of money, in dollars, raised from ticket sales as a function of ?

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SAT Linear Functions Question 81 | Free SAT Prep | Aniko

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Question 81·Hard·Linear Functions

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At a charity festival, the first 50 admission tickets sold raise $12 each for the organization. Every additional ticket sold beyond the first 50 raises $18 each. Let $t$ represent the total number of tickets sold, where $t \ge 50$ . Which function $R$ gives the total amount of money, in dollars, raised from ticket sales as a function of $t$ ?

For linear modeling questions with different rates before and after a cutoff (like “the first 50 tickets cost this much, the rest cost that much”), first compute the fixed amount from the initial group (here, $50\cdot 12$ ). Then express the number of additional items as $(\text{total} - \text{initial count})$ and multiply by the second rate (here, $18(t-50)$ ). Add these two parts, simplify the expression, and quickly verify your answer by plugging in the cutoff value (like $t=50$ ) to check it matches the situation described.

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Start with the first 50 tickets

What is the total amount of money raised if exactly 50 tickets are sold, with each of those tickets raising $12 each for the organization?

Count only the extra tickets for the $18 rate

If $t$ is the total number of tickets sold, how many of those tickets are beyond the first 50? Write that number in terms of $t$ .

Build the total revenue expression

Write the total revenue as (money from the first 50 tickets) plus (money from each extra ticket times the number of extra tickets). Then simplify your expression and compare it to the answer choices.

Desmos Guide

Compute the revenue at 50 tickets

In Desmos, type 50*12 to find the exact revenue when exactly 50 tickets are sold. Note this value; the correct function must give this same output when the input is 50.

Enter each answer choice as a function

On separate lines, enter y=18x-300, y=18x+300, y=12x+600, and y=18x+900 in Desmos (use $x$ in place of $t$ ). This graphs all four candidate functions.

Check which function matches the known revenue at 50 tickets

For each graph, either click on the point where $x=50$ or type the expression with 50 substituted for $x$ (for example, 18*50-300). Compare each y-value to the number you got from 50*12; the function whose value matches is the correct revenue function.

Step-by-step Explanation

Find the revenue from the first 50 tickets

Since $t \ge 50$ , there are always at least 50 tickets sold. The first 50 tickets raise $12 each, so the revenue from these is

50 \cdot 12 = 600

This $600 is a fixed amount that does not change when $t$ changes.

Express the revenue from the additional tickets

Any tickets beyond the first 50 raise $18 each.

If $t$ is the total number of tickets sold, then the number of additional tickets is $t-50$ .
Each of those additional tickets brings in $18.

So the revenue from the additional tickets is $18(t-50)$ dollars.

Write an expression for the total revenue

Total revenue is the sum of:

$600 from the first 50 tickets, and
$18(t-50)$ dollars from the additional tickets.

So an expression for the total revenue in terms of $t$ is

$600 + 18(t-50)$ .

Simplify and match to the choices

Now simplify the expression $600+18(t-50)$ :

600 + 18(t-50) = 600 + 18t - 900 = 18t - 300

Therefore, the revenue function is $R(t)=18t-300$ , which corresponds to choice A.

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Step-by-step Explanation

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Step-by-step Explanation