MODELING USING LINEAR FUNCTIONS

Problem 1 :

Sixty students on ‘The Great Bike Ride’ consume either three meals a day or four meals a day. If 213 meals are consumed on one particular day, how many students ate four meals?

Solution:

Let t = students having three meals in a day

and f = students having four meals in a day

t + f = 60 ---> (1)

3t + 4f = 213 ---> (2)

(1) × 3 ==>  3t + 3f = 180 ---> (3)

(2) – (3):

f = 33

33 students have four meals on that day.

Problem 2 :

Petra is four times as old as her daughter Josie.In four years time she will be only three times as old as Josie. How old is Petra and Josie?

Solution:

P = 4J

P + 4 = 3(J + 4)

4J + 4 = 3(J + 4)

4J + 4 = 3J + 12

J = 8

P = 4 × 8

P = 32

Petra is 32 years old and Josie is 8 years old.

Problem 3 :

What is the equation of the line that passes through the points (-1, 0) and (3, -2)?

Solution:

The two points are (-1, 0) and (3, -2).

The equation of the line

y - y₁ = m(x - x₁)

Substituting (-1, 0) and m = -1/2.

Problem 4 :

David delivers pamphlets and is paid $25 to make the deliveries plus 20 cents per pamphlet delivered.

a) How much will Davis be paid if he delivers 1750 pamphlets?

b) How many pamphlets would David need to deliver to earn $500?

Solution:

(a)

Let P = total amount paid

n = number of pamphlets delivered.

P = 25 + 0.2n

n = 1750

P = 25 + 0.2 × 1750

P = 25 + 350

P = 375

David is paid $375.

(b)

P = 25 + 0.2n

500 = 25 + 0.2n

475 = 0.2n

David will need to deliver 2375 pamphlets.

Problem 5 :

A theme park charges $25 for adults and $15 for children. A party of 21 people pay $375 altogether. How many adults and how many children were there?

Solution:

Let the number of adults = A

The number of children = C

25A + 15C = 375 ---> (1)

A + C = 21 ---> (2)

C = 21 - A

Substitute C = 21 - A in (1)

25A + 15(21 - A) = 375

25A + 315 - 15A = 375

10A = 60

A = 6

Substitute A = 6.

C = 21 - 6

C = 15

6 adults and 15 children went to the theme park.

Problem 6 :

The materials to make 25 kg of an alloy of copper and zinc cost $62. If copper costs $3.20 per kg and zinc costs $1.40 per kg, find the composition of the alloy.

Solution:

3.2C + 1.4Z = 62

C + Z = 25

Z = 25 - C

3.2C + 1.4(25 - C) = 62

3.2C + 35 - 1.4C = 62

1.8C = 27

C = 15

15 + Z = 25

Z = 10

There is 10 kg of zinc and 15 kg of copper.