WORD PROBLEMS INVOLVING LINEAR MODEL

Problem 1 :

The cost of a school banquet is $95 plus $15 for each person attending. Write an equation that gives total cost as a function of the number of people attending. What is the cost for 77 people?

Solution :

Let x be the number of persons attending the banquet.

Total cost of school banquet  =  95 + 15x

Number of people attending(x)  =  77

Cost of attending 77 people  =  95+15(77)

=  95 + 1155

=  $1250

So, the total cost of attending 77 people is $1250.

Problem 2 :

In 1980 the average price of a home in Brainerd County was $97,000. By 1986 the average price of a home was $109,000. Write a linear model for the price of a home, P, in Brainerd County as a function of the year, t. Let t = 0 correspond to 1980.

Solution :

In the year of 1980, the average price of home is

(1980, 97000) and (1986, 109000)

Rate of change  =  (109000 - 97000)/(1986-1980)

=  12000/6

=  2000

Let "P" be the price of home.

Equation of linear model :

y = mx + b

P  =  2000t + 97000

Problem 3 :

Roman paid $150 to join a handball club. He pays an additional $15 every time he uses one of the club's handball courts. Write an equation that describes Roman's total cost for playing handball as a function of the number of times he plays.

Let C = the total cost and n = the number of times he plays.

Solution :

150  =  fixed charge and 15 number of times he plays

Given : n be the number of times he plays

C  =  150 + 15n

Problem 4 :

A sunflower in Julia Rosario's garden was 12 centimeters tall when it was first planted. Since then, it has grown approximately 0.6 centimeters per day. Write an equation expressing the sunflower's height, H, in terms of the number of days, d, since it was planted.

Solution :

Height of sunflower  =  H

Growing at the rate of 0.6 centimeter per day.

Let d be the number of days it was planted.

H  =  12 + 0.6d

Problem 5 :

Billy plans to paint baskets. The paint costs $14.50. The baskets cost $7.25 each. Write an equation that gives the total cost as a function of the number of baskets made. Determine the cost of four baskets.

Solution :

Cost of paint  =  $14.50

Cost of basket  =  $7.25

Let x be the number of baskets made.

Total cost  =  14.50 + 7.25x

Cost of four baskets :

x  =  4

Total cost  =  14.50 + 7.25(4)

=  14.50 + 29

=  $43.5 

Problem 6 :

A real estate sales agent receives a salary of $250 per week plus a commission of 2% of sales. Write an equation that gives the weekly income y in terms of sales x.

Solution :

Given :

Weekly income  =  y, x  =  sales

Salary  =  $250 per week

Commission  =  2% of sales

=  0.02x

y  =  250+0.02x

Problem 7 :

If a large factory sells its new gadgets for $5 each, it can sell 1050 per month, and if it sells the same gadgets for $9, it will sell 900 per month. Assuming the relationship between price and sales is linear, predict the monthly sales of gadgets to the nearest whole number if the price is $12.

Solution :

Given :

The relationship between price and sales is linear.

y  =  mx+b

1050  =  5x + b   ----(1)

900  =  9x + b  ----(2)

(1) - (2)

5x + b  =  1050

9x + b  =  900

(-)   (-)     (-)

---------------

-4x  =  150

x  =  -37.5

By applying the value of x in (1), we get

5(-37.5) + b  =  1050

-187.5 + b  =  1050

b  =  1050+187.5

b  =  1237.5

y  =  -37.5m + 1237.5

y  =  -37.5 (12) + 1237.5

y  =  -450+1237.5

y  =  787.5

y  =  788

Monthly sales of gadgets is $788, when the price of one gadget is $12.

Problem 8 :

The graph for a stable that charges a $20 flat fee plus $10 per hour for horseback riding is shown below.

How will the graph change if the stable changes its charges to a flat fee of $45 plus $30 per hour?

Solution :

y-intercept of the graph is 20 and slope is 10.

New flat fee is $45 and 30 per hour. 

y-intercept  =  $45 and rate of change  =  $30.

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