In a photography studio, the total charge , in dollars, for renting a set of lighting equipment is given by the formula above, where is the number of hours the equipment is rented. For a special promotion, the studio requires that the total charge be at least $95 and at most $140.
Which of the following inequalities gives the possible values of that satisfy this requirement?
A compound inequality combines a lower and an upper limit on the same quantity. For a rental charge, translate the dollar limits first. Then graph the charge formula in Desmos and find where it reaches each limit. Because the charge rises with time, the hours between those crossings work. Don't mistake dollar amounts for hours.
Hints
- Hint 1
The phrases at least and at most both include their boundary values. A compound inequality puts the total charge between two amounts at once. What belongs on either side of ?
- Hint 2
Graph the charge against hours. At an intersection with a horizontal dollar-limit line, both graphs show the same charge. Which coordinate of that point tells you the number of hours?
- Hint 3
Desmos may round the upper hour limit on the graph. To get an exact fraction, subtract the fixed charge from , then divide by the hourly rate. Why does that give hours?
Step-by-step
Graph the two charge limits
Step 1Translate the charge requirement
The charge must be at least $95 and at most $140. Both limits count, so put the charge between them in a compound inequality, which states two limits at once:
- Step 2
Find the lower hour limit
Graph , using for hours and for dollars. Add to mark the lower charge limit. Click their intersection, where the charges match. Desmos shows , so the charge first reaches $95 at hours.
- Step 3
Find the upper hour limit
Add for the upper charge limit. Click where it meets the charge line. Desmos shows about , so the charge reaches $140 at about hours. That graph reading is rounded; the choices use an exact fraction.
- Step 4
Get the exact upper limit
To find the exact number of hours, remove the fixed $40 from $140, then divide by the $5.5 charged per hour. Type in Desmos. It displays ; its fraction button shows .
- Step 5
Write the allowed rental times
The charge rises by $5.5 each hour, so the hours between the two crossings meet both charge limits. The limits are included, so
Convert dollar limits to hour limits before writing the range. These are the possible rental times. Choice D.