A stall sells (11+x) items at ($21−x) each and pays a fixed $2 cost. For which two whole-number values of x is its profit $253?
Show worked answer
Answer: x=4x=6
Profit is (11+x)(21−x)−2.
Equate it to 253 and rearrange: x2−10x+24=0=(x−4)(x−6).
Both 4 and 6 give positive prices and item counts.
Example 2
A rectangular garden is 7m longer than it is wide. A path 3m wide surrounds all four sides. Garden and path together cover 170m2. Find the perimeter of the garden itself.
Garden and surrounding pathDiagram not to scaleShow worked answer
Answer: 30
Let garden width be x.
The outside dimensions are x+6 and x+13.
Solve their product =170;
the positive solution is x=4.
Garden length is 11
its perimeter is 2(4+11)=30m.
Example 3
A stall sells (9+x) items at ($17−x) each and pays a fixed $2 cost. For which two whole-number values of x is its profit $158?
Show worked answer
Answer: x=1x=7
Profit is (9+x)(17−x)−2.
Equate it to 158 and rearrange: x2−8x+7=0=(x−1)(x−7).
Both 1 and 7 give positive prices and item counts.
Quadratic Modelling II practice | Year 10 | Ascent Arcade