The applied problems in this section are similar to problems earlier in this chapter. The only difference is that you are given a range for one value and you are asked to find the range for the other.
Examples
A high
school student earns $8 per hour in her summer job. She hopes to earn between
$120 and $200 per week. What range of hours will she need to work so that her
pay is in this range?
Let x represent the number of hours worked
per week. Represent her weekly pay by p = 8x. The student wants 120 ≤ p ≤ 200.
The inequality to solve is 120 ≤ 8x ≤ 200.
The student
would need to work between 15 and 25 hours per week for her pay to range from
$120 to $200 per week.
A
manufacturing plant produces pencils. It has monthly overhead costs of $60,000.
Each gross (144) of pencils costs $3.60 to manufacture. The company wants to
keep total costs between $96,000 and $150,000 per month. How many gross of
pencils should the plant produce to keep its costs in this range?
Let x represent the number of gross of
pencils manufactured monthly. Production cost is represented by 3.60x.
Represent the total cost by c = 60,000 + 3.60x. The manufacturer wants 96,000 ≤
c ≤ 150,000. The inequality to be solved is 96,000 ≤ 60,000 + 3.60x ≤ 150,000.
The
manufacturing plant should produce between 10,000 and 25,000 gross per month to
keep its monthly costs between $96,000 and $150,000.
Practice
1. According
to Hooke’s Law, the force, F (in pounds), required to stretch a certain spring x
inches beyond its natural length is F = 4.2x. If 7 ≤ F ≤ 14, what is the
corresponding range for x?
2. Recall
that the relationship between the Fahrenheit and Celsius temperature scales is
given by F = 9/5 C + 32. If 5 ≤ F ≤ 23, what is the
corresponding range for C?
3. A
saleswoman’s salary is a combination of an annual base salary of $15,000 plus a
10% commission on sales. What level of sales does she need to maintain in order
that her annual salary range from $25,000 to $40,000?
4. The
Smith’s electric bills consist of a base charge of $20 plus 6 cents per
kilowatt-hour. If the Smiths want to keep their electric bill in the $80 to
$110 range, what range of kilowatt-hours do they need to maintain?
5. A
particular collect call costs $2.10 plus 75 cents per minute. (The company
bills in two-second intervals.) How many minutes would a call need to last to
keep a charge between $4.50 and $6.00?
Solutions
If the force is to be kept between 7
and 14 pounds, the spring will stretch between 5/3 and 10/3 inches beyond its
natural length.
3. Let
x represent the saleswoman’s annual sales. Let s = 15,000 + 0.10x represent her
annual salary. She wants 25,000 ≤ s ≤ 40,000.
She needs to have her annual sales range from $100,000 to $250,000 in
order to maintain her annual salary between $25,000 and $40,000.
4. Let
x represent the number of kilowatt-hours the Smiths use per month. Then c = 20 +
0.06x represents their monthly electric bill.
The Smiths would need to keep their monthly usage between 1000 and 1500
kilowatt-hours monthly to keep their monthly bills in the $80 to $110 range.
5. Let
x represent the number of minutes the call lasts. Let c = 2.10 + 0.75x represent
the total cost of the call.
3.2 minutes is three minutes 12 seconds and 5.2 minutes is five minutes
12 seconds because 0.20 minutes is 0.20(60) seconds = 12 seconds
A
call would need to last between three minutes 12 seconds and 5 minutes 12
seconds in order to cost between $4.50 and $6.00.
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