Problem 1 :
A long distance telephone charges 99 cents for any call upto 20 minutes in length and 7 cents for each additional minute. Use bracket notation to write a formula for the cost C, of a call as function of its length time t in minutes. Graph the function. How much does it cost to talk for
i) 10 minutes ?
ii) 25 minutes ?
Solution :
Charges collected for long distance call = 0.99
For first 20 minutes (≤)
Let t be the additional minutes.
Charge for each additional minute = 0.07
Charge after t minute = 0.99 + 0.07t when t > 20
i) 10 minutes
t = 10 < 20
C(t) = 0.99
ii) 25 minutes
t = 25 > 20
C(t) = 0.99 + 0.07t
Applying t = 25 in the function above
C(25) = 0.99 + 0.07(25)
= 0.99 + 1.75
= 2.74
Problem 2 :
Suppose a carpet store sells carpet for $10 per square yard for the 100 square yards purchased, and then lower the price to $7 per square yard after the first 100 yards have been purchased. Find the function C = f(x), that gives the cost of purchasing any number of square yards of carpet between 0 and 200 square yards. How much does it cost for
i) 50 square yards ?
ii) 150 square yards ?
Solution :
Price of carpet per square yard = $10
for 100 square yards purchased
Let x be the area of the carpet purchased.
Cost spent for 100 yards = 100 (10) ==> 1000
After 100 yards, the cost spent = 0.07 (100 - x)
Cost spent for x area of the carpet = 1000 + 0.07 (100 - x)
for 100 ≤ x ≤ 200
Problem 3 :
On a trip, the total distance (in miles) you travel in x hours is represented by a piecewise function
How far do you travel in 4 hours ?
Solution :
When x = 4, d(x) = 65x - 20
Applying the value of x, we get
d(4) = 65(4) - 20
= 260 - 20
= 240 miles
Problem 4 :
The total cost (in dollars) of ordering x custom shirts is represented by the piecewise function.
Determine the total cost of ordering 26 shirts.
Solution :
When x = 26
c(x) = 15.80x + 20
Applying the value of x, we get
c(26) = 15.80 (26) + 20
= 410.8 + 20
= 430.8
Problem 5 :
A car company charges $45 plus $0.20 per mile over 50 miles.
Give the equation for the cost of driving 0 ≤ m ≤ 50 miles
Give an equation to determine the cost of driving more than 50 miles (m > 50)
i) How much will it cost you if you drive 20 miles ?
ii) How much will it cost you if you drive 51 miles ?
Solution :
Let m be the number of miles driven.
Cost of driving 0 ≤ m ≤ 50 miles :
c(m) = 45
Cost of driving m > 50 miles :
c(m) = 45 + 0.20(m - 50)
i) When the distance to be covered = 20 miles.
This distance lies between 0 to 50, so the required cost is $45.
ii) When the distance to be covered = 51 miles.
c(m) = 45 + 0.20(m - 50)
= 45 + 0.20(51- 50)
= 45 + 0.20(1)
= $45.20
Problem 6 :
A cell phone company charges a monthly fee $9.95, and a usage fee as follows.
i) Write a piecewise function, C(m) for the cost of using m minutes.
ii) What is the total cost, if you use 200 minutes ?
iii) What is the total cost, if you use 350 minutes ?
Solution :
i) Let m be the number of minutes.
Monthly fee :
When m < 150 When 150 ≤ m ≤ 300 When m > 300 |
9.95 plus 0.40 9.95 plus 0.20(m - 150) 9.95 plus 0.10(m - 300) |
ii) What is the total cost, if you use 200 minutes ?
c(200) = 9.95 + 0.20(200 - 150)
= 9.95 + 0.20(50)
= $19.95
iii) What is the total cost, if you use 350 minutes ?
c(350) = 9.95 + 0.10(350 - 300)
= 9.95 + 0.10(50)
= $24.95
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