A theater sells adult tickets for $15 each and student tickets for $10 each. For an evening show, the theater must sell at least 50 tickets, and the total revenue from ticket sales can be no more than $700.
What is the maximum number of adult tickets the theater can sell for that show?
Words like “at least” and “no more than” signal a system of inequalities: both limits must hold at once. Let the horizontal coordinate count what you're maximizing, graph each limit in Desmos, and look for the rightmost point in the overlapping shading. Don't use the revenue limit alone; a combination under the cap might sell too few tickets.
Hints
- Hint 1
An inequality allows a total to be above or below a limit, not only equal to it. Let count adult tickets and count student tickets. What inequality says the combined count is at least the required minimum?
- Hint 2
Each adult ticket adds $15 to revenue, and each student ticket adds $10. The theater can collect no more than its revenue cap. How would you write that second inequality?
- Hint 3
On a graph, larger adult counts lie farther right. The overlap contains pairs that meet both limits, so look for its rightmost point, not the right edge of either shading by itself.
Step-by-step
Approach 1: Graph the limits in Desmos
Step 1Turn the ticket minimum into an inequality
Let be adult tickets and be student tickets. Their total is , so at least 50 tickets means . Type that into Desmos with and , since ticket counts can't be negative. The shading shows pairs that meet the ticket minimum.
- Step 2
Add the revenue limit
Adult tickets bring in dollars and student tickets bring in dollars. No more than $700 means . Add it to Desmos. The overlapping shading now shows pairs that meet both limits.
- Step 3
Choose the edge that can give the most adults
Increasing moves you right. For a fixed adult count, extra student tickets only raise revenue, so they can't help you stay under the cap. For a maximum, use the farthest-right point that satisfies every limit. Look for the rightmost corner of the overlap, where the ticket-minimum and revenue-limit lines meet.
- Step 4
Read and check the rightmost point
Click that corner. Desmos shows , meaning 40 adult tickets and 10 student tickets. Their count is . Type ; Desmos prints . Both limits allow equality, and the graph has no overlap farther right. So the maximum is 40 adult tickets. Choice C.
Approach 2: Start with student tickets and make replacements
Step 1Find the cost of the cheapest required 50 tickets
To leave the most room under the revenue cap, start with exactly 50 tickets. Making all 50 student tickets gives the cheapest starting total. Type ; Desmos prints . Selling extra tickets instead would add revenue, not make room for more adults.
- Step 2
Find the cost of each adult replacement
Replace one student ticket with one adult ticket. The total stays at 50, but revenue rises by the price difference. Type ; Desmos prints . Each replacement uses $5 more of the revenue allowance.
- Step 3
Use the revenue cap to limit replacements
If is the number of replacements, revenue is . Type to find where it reaches the $700 cap. Desmos reports under PARAMETERS. That leaves 10 student tickets, so 40 adults meet both limits; one more replacement would exceed the cap. Choice C.