A quadratic profit model written in vertex form shows where its graph turns. A negative coefficient on the square makes that turning point a maximum. Check that its input is allowed, then graph the model in Desmos and click the highest point. The horizontal coordinate tells you how many units to produce; the vertical coordinate gives the profit.
Hints
- Hint 1
A square can never be negative. The model subtracts , so when is that subtraction zero? That input gives the largest possible output.
- Hint 2
The domain is the set of allowed inputs. Check whether the input you found lies between the stated endpoints; a peak outside that interval wouldn't be a possible profit.
- Hint 3
The vertex is the parabola's turning point. Desmos can label it as a point. Which coordinate gives dollars of profit, rather than hundreds of units?
Step-by-step
Find the allowed peak
Step 1Find where the profit peaks
A square can't be negative, so . This term is zero when and negative everywhere else. So gives the highest point, called the vertex, of this profit graph. The multiplies the square; it isn't a profit amount.
- Step 2
Check that the peak is possible
The domain means the inputs the model allows. Since , the peak at is within the allowed interval. Here counts hundreds of units, so that input means units.
- Step 3
Read the maximum profit
Type in Desmos and click the vertex. Desmos labels it . The first coordinate counts hundreds of units; the second coordinate is the profit, in dollars. So the maximum possible daily profit is $392. Choice D.