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MCR3U – QUADRATIC WORD PROBLEMS
1.
A community recreation centre wants to enclose a rectangular swimming area at a public beach. The city has provided 300m of rope
and buoys to mark off the area. What is the maximum swimming area that can be enclosed along the beach with the 300m of rope?
What are the dimensions of the swimming area? [11250m², 150mx75m]
2.
A glassworks that makes lead-crystal bowls has a daily production cost C in dollars given by C = 0.2b - 10b + 650, where b is the
number of bowls made. How many bowls should be made to minimize the production cost? What is the cost when this many bowls
are made? [25, $525]
3.
The hypotenuse of a right triangle is 26cm. The sum of the other two sides is 34cm. Determine the length of the other two sides.
[10,24]
4.
A rectangular field along a straight river is to be divided into 3 smaller fields by one fence parallel to the river and 4 fences perpendicular to the
2
river. Find the maximum area that can be enclosed if 1600m of fencing is used. [160000m ]
5.
Suppose a stream borders our land, and we want to make a right triangular garden with the stream as the hypotenuse. If we have only 80 meters
2
of fencing, what is the maximum area of our garden? [800m ]
6.
The city transit system carries 24 800 bus riders per day for a fare of $1.85. A survey indicates that the number of riders will increase
by 800 for every $0.05 decrease in the fare. What fare(s) will produce a revenue of $46 080? [$1.60 or $1.80]
7.
The sum of two numbers is 26. The sum of their squares is a minimum. Determine the numbers. [13,13]
8.
A rectangle is 3cm longer than it is wide. The diagonal is 15cm. Determine the dimensions of the rectangle. [12cm x 9cm]
9.
Determine two numbers whose difference is 13 and whose squares when added together yield a minimum.
2
[-6.5,6.5]
10. The altitude of a triangle is 2m longer than its base. What are the dimensions of the altitude and the base if the area of the triangle is
2
40 m ? [b=8m, h=10m]
11. At a park, a landscaper wishes to plant a uniform border of tulips on the inside of a rectangular garden with dimensions of 18m by
12m. The garden will look best if the area of the tulip border is half the area of the garden. How wide should the inside of the border
of tulips be, to one decimal place? [2.1m]
2
12. A football is punted into the air. Its height, h, in metres after t seconds is given by h(t) = -4t + 24.5t + 1 Find the maximum height of
the ball. (Round to the nearest hundredth of a metre.) When does the ball reach its maximum height? [38.52m,3.06s]
13. Slacks Incorporated sold 6000 pairs of pants last month at an average price of $84 each. The store is going to increase prices in order to increase
profits. Sales forecasts indicate that sales will drop by 100 for every dollar increase in price. On average, the company pays $37 for each pair of
pants it sells. How many pairs of pants must they sell in order to maximize profit? [5350]
14. An auto parts store currently sells 300 spark plug packages each week at a price of $6.40 each. To increase sales and reach more customers the
parts outlet decides to reduce the price of the package, knowing that every 10-cent decrease in price will result in 5 more sales. What price will
maximize the total revenue? [$6.20]
15. The sum of the squares of three consecutive integers is 2189. Find the integers. [26,27,28 or -28,-27,-26]
16. A rectangular sheet of tin is 15cm long and 9cm wide. A uniform strip is to be off around the sheet. The remaining area is to be
112cm 2 . Calculate the width of the strip. [0.5cm]
17. A resort hotel has 250 rooms and the standard rate is $150 per night per room. During the non-holiday season 100 rooms are rented
each day on average but for each $20 price reduction, 25 more rooms are rented. What non-holiday price would maximize revenue?
[$115]
18. A company is selling DVD players for $150. They sell 60 each day. The company is looking to increase the selling price in order to increase profits.
A survey indicates that for each $2.00 increase, one customer will be lost. The cost of making the DVDs is $95.00. How much should the company
charge to maximize profits? [$182.50]
19. What is the maximum area of a triangle having 15 cm as the sum of the base and height? [28.125cm²]
20. Maria produces and sells shell necklaces. The material fore each necklace costs her $4. She has been selling them for $8 each and averaging sales
of 40 per week. She has been told that she could charge more but has found that for each $0.50 increase in price, she would lose 4 sales each
week. To maximize profit, what selling price should she set and what would her profit per week be at this price? [$8.50,$162]
21. What number exceeds its square by the greatest amount? [0.5]
22. A sportswear store sells baseball caps with the local baseball team’s logo on them. Last year, the store sold 600 caps at $15 each. The store
manager is planning to increase the price. A consumer survey shows that for every $1 increase, there will be a drop of 30 sales a year.
a) State the revenue equation, R x  . [ (600  30x )(15  x ) ]
b)
c)
d)
e)
f)
g)
If the overhead costs are $8.00 per cap, determine the profit equation, P x  . [ (600  30x )(15  x )  8(600  30x ) ]
What should the selling price be to maximize the annual revenue? What is the maximum revenue? [$17.50, $9187.50]
What selling price will maximize the profit? [$21.50]
What is the maximum profit? [$5467.50]
How many baseball caps must they sell to break even? [none or 810]
What selling price will ensure them a profit of $5280? [$19 or $24]