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Math 11 Applications
Outcomes D6 and D7
APPLICATIONS WITH SUBSTITUTION
Example 1:
A rescue helicopter flies at 3 miles a minute. When the helicopter is 22 miles away a boat sends up
a flare. The helicopter heads toward the boat. An equation for the distance d in miles away from
the boat at time t minutes is d = -3t +22. The helicopter can spot a flare 6 miles away. The equation
for the distance when the flare is visible is d = 6. How long until the helicopter pilot sees the flare?
Example 2:
A store receives an order of 300 TV sets in two deliveries. The first delivery has 60 more sets than
the second delivery. The equations are:
f + s = 300 and f = s + 60, where f is the number in the first delivery and s is the number in the
second delivery. How many in each delivery?
Math 11 Applications
Outcomes D6 and D7
Example 3:
The mass of wheat a farmer can supply is given by the equation W = 4000p – 3000
The mass of wheat that is bought (demand) is given by the equation W = -3000p + 25000
Where p is the selling price per bag and W is the mass of wheat.
Determine the price per bag if supply equals demand
Practice: For each question show the substitution and the working to obtain the answer.
1. A rescue helicopter flies at 5 miles a minute. When the helicopter is 35 miles away a boat sends
up a flare. The helicopter heads toward the boat. An equation for the distance d in miles away from
the boat at time t minutes is d = -5t +35. The helicopter can spot a flare 10 miles away. The
equation for the distance when the flare is visible is d = 10. How long until the helicopter pilot sees
the flare?1
2. A store receives an order of 1000 stereos in two deliveries. The first delivery has 250 more sets
than the second delivery. The equations are:
f + s = 1000 and f = s + 250, where f is the number in the first delivery and s is the number in the
second delivery. How many in each delivery?2
1
2
10 hours
625 stereos in the first delivery and 375 stereos in the second delivery
Math 11 Applications
Outcomes D6 and D7
3. The mass of wheat a farmer can supply is given by the equation W = 3500p – 2000
The mass of wheat that is bought (demand) is given by the equation W = -4000p + 30000
Where p is the selling price per bag and W is the mass of wheat.
Determine the price per bag if supply equals demand.3
4. An airplane cruises at a constant velocity of 840 km/h during the 5430-km trip from Montreal to
London, England. The equation that describes its motion is d = -840t + 5430, where d is its distance
from London in kilometers, and t is the time in hours since the airplane reached its cruising
velocity. The plane must start its descent when it is 90 km from London, do d = 90. How long has
the plane been cruising when it starts its descent?4
5. A car rental agency offers two rental plans for a certain class of car. Plan 1 is $60 per day with
no charge for the number of kilometers driven. Plan 2 is $20 per day, plus $0.25 for each kilometer
driven. You wish to rent a car for one day. The daily cost , c dollar, fro driving d kilometers is:
Plan 1: C = 60
Plan 2: C = 0.25d + 20
a) Which plan is better for a one-day, 200-km trip?
b) Solve the linear system. Determine how far you could drive so that the rental costs for each plan
would be the same. What is the rental cost?5
3
$4.27/bag
Approximately 6.36 h
5
a) Plan 1
b) 160 km; $60
4
Math 11 Applications
Outcomes D6 and D7
6. Pyramid Stable scharges $20/h (including insurance) for trail rides. Sara’s Stables charges
$16/h, with a separate fee of $12. Let the cost of the trail rides be C dollars and the number of
hours be h. The equations relating the cost to hours for each stable are C = 20h and C = 16h + 12.
a) How many hours of trail rides would result in the same total costs for each stable?
b) Suppose you wish to go riding for 2 h. Which stable would you choose? Justify your answer.6
7. Ideally, the quantity of a crop that farmers harvest and sell at a given price should equal the
quantity of the crop that consumers are willing to buy at that price. This is the law of supply and
demand. The price of grapes is at least $1/kg and never more than $6.50/kg. The mass of grapes, G
kilograms, that farmers will harvest (supply) is represented by G = 5000p − 5000, where p dollars is
the selling price per kilogram. The quantity of grapes that consumers will purchase (demand) is
represented by G = -4000p + 26 000. Determine the price per kilogram where the supply of grapes
equals the demand.7
6
7
a) 3 h
b) Pyramid Stables; answers will vary but work must support your answer
Approximately $3.44/kg