Question 402·Hard·Algebra

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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 $t$ ?

The cue is two ticket rates separated by a cutoff. For the allowed ticket totals, revenue follows a linear model: after the cutoff, each new ticket adds the same amount. Subtract the first block from the total before applying the later rate, then add the two groups' earnings. Check the rule at the cutoff in Desmos; treating every ticket as a later ticket overcounts the first block.

Hints

Hint 1
The variable counts all tickets, but only tickets beyond the first block earn the higher amount. To find that second count, what must you subtract from $t$ ?
Hint 2
A rate is the dollars raised per ticket. Multiply each group's rate by its ticket count, then add the two dollar amounts. How many tickets belong in each group?
Hint 3
The distributive property lets you multiply the later rate by both terms in its ticket count. What happens when you distribute across the subtraction?

Step-by-step

Count each ticket group
Step 1
Count the later tickets
Since $t$ counts all tickets, subtract the first $50$ to count only those sold beyond them: $t-50$ . The domain, meaning the allowed inputs, says $t\ge 50$ , so this count can't be negative.
Step 2
Build the revenue rule
Multiply each rate, in dollars per ticket, by its ticket count. The first $50$ tickets raise $12(50)$ dollars, and the later $t-50$ tickets raise $18(t-50)$ dollars. Add the two amounts: $R(t)=12(50)+18(t-50)$ . Type $f(t)=12(50)+18(t-50)$ in Desmos to name this rule $f$ there.
Step 3
Check the cutoff
Type $f(50)$ , meaning put $50$ in for the total ticket count. Desmos prints $600$ . At exactly $50$ tickets, none has earned the later rate, so $600$ must be the money from the first block alone.
Step 4
Distribute the later rate
Use the distributive property: $18$ multiplies both $t$ and $-50$ . Distribute it: $R(t)=12(50)+18t-18(50)$ . The subtraction removes the first $50$ tickets from the group charged at the later rate.
Step 5
Combine the fixed amounts
Type $12(50)-18(50)$ in Desmos. It prints $-300$ . Replace those fixed amounts with that value: $R(t)=18t-300$ . The negative constant term corrects for the first tickets; it doesn't mean the organization loses money. The rule gives the total dollars raised for $t\ge 50$ . Choice D.

Question 402·Hard·Algebra

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