Write an inequality for the problem and determine the constrains on the variables. Then use your inequality to answer the question.
Reza and James have started exercising together. When they started Reza weighed 185 pounds and James weighed 180 pounds . After 10 weeks, Reza
weighed less than James. Assume that the weight change for each man was about the same , on average, each week.
Nell sells pies each at a farmer's market. She spends $65 on materials and charges $10 per pie. She hopes to make a profit of at least $15 per hour. What constraints might Nell want to set?
Let's assume that Reza was losing x pounds per week and James was losing y pounds per week. Then, after 10 weeks, Reza would weigh 185 - 10x pounds and James would weigh 18 - 10y pounds.
If after 10 weeks Reza weighed less than James, the inequality to describe the situation would be
185 - 10x < 180 - 10y
To determine the constraints on the variables x and y, combine the variable terms on the right side and the numbers on the left side:
185 - 180 < 10x - 10y
5 < 10(x - y)
Dividing both sides by 0 yields
x - y > 0.5 or x > y + 0.5
This means, that in order for Reza to weigh less than James after 10 weeks, her weight loss per week has to be by half a pound greater than James's weight loss per week.
2) Denote the number of pies Nell is going to sell by p. Then, she is going to make $10p by selling p pies, since she charges $10 per pie. Since she spends $65 on materials, her net profit will be, in dollars, 10p - 65.
If Nell is going to spend t hours at the market, and wants to make at least $15 per hour, then she wants her net profit to be at least $15t.
The inequality that describes the situation is
`10p - 65 >=15t`
`10p - 15t >=65`
Dividing both sides by 5 yields
`2p - 3t >=13`
Nell could use the above inequality to set constraints on the number of pies she should make. For example, if she wants to spend t = 3 hours at the market, she would have to make and sell at least p pies, where p is
`2p - 3*3 >=13`
` ` `2p>=22`
If Nell wants to spend 3 hours at the market, she would have to make and sell at least 11 pies, in order to have profit of at least $15 an hour.
Alternatively, if Nell could set constraints on the time spent at the market. If, for example, she had make p = 16 pies, the time she should spend at the market is determined by
`2*16- 3t >=13`
Nell would have to spend no more than 3 hours at the market selling 16 pies, if she wants to make at least $15 per hour.