Question

jeremy and his dad are sharing the responsibilities for their new plumbing company. combined, jeremy and his dad worked at least 85 combined hours last week. they collected a total of more than $1000 by charging $20 per hour. let x = jeremy’s hours, let y = his dad’s hours, and let this system represent the situation: ( x + y geq 85 ); ( 20(x + y) > 1000 ); ( x geq 0 ); ( y geq 0 ). graph with overlapping region labeled overlapping region. a. describe what the constraints mean. b. identify two solutions and explain what they each mean.

Explanation:

Part a: Describe what the constraints mean

for \( x + y \geq 65 \):
The variable \( x \) represents Jeremy's hours, and \( y \) represents his dad's hours. The inequality \( x + y \geq 65 \) means that the total number of hours Jeremy and his dad worked combined is at least 65 hours last week.

for \( 20(x + y) > 1000 \):
Since they charge $20 per hour, the total money earned is LXI0 times the total hours \( (x + y) \). The inequality \( 20(x + y) > 1000 \) (which can be simplified to LXI1 by dividing both sides by 20) means that the total money they collected is more than $1000.

for \( x\geq0 \) and \( y\geq0 \):
The inequalities \( x\geq0 \) and \( y\geq0 \) mean that the number of hours Jeremy worked (\( x \)) and the number of hours his dad worked (\( y \)) cannot be negative, as you can't work a negative number of hours.

Part b: Identify two solutions and explain what they each mean

A solution is a pair of non - negative values \( (x,y) \) that satisfy all the inequalities \( x + y\geq65 \), \( 20(x + y)>1000 \), \( x\geq0 \), and \( y\geq0 \).

Solution 1: \( (x = 30,y = 35) \)

• Step 1: Check \( x + y\): \( 30+35 = 65 \), and \( 65\geq65 \), so it satisfies \( x + y\geq65 \).
• Step 2: Check \( 20(x + y) \): \( 20\times(30 + 35)=20\times65 = 1300 \), and \( 1300>1000 \), so it satisfies \( 20(x + y)>1000 \).
• Step 3: Check \( x\geq0 \) and \( y\geq0 \): \( 30\geq0 \) and \( 35\geq0 \).

Meaning: Jeremy worked 30 hours and his dad worked 35 hours. Their total hours are 65 (which meets the minimum - hour requirement), and their total earnings are \( 20\times65 = 1300 \) dollars, which is more than $1000.

Solution 2: \( (x = 40,y = 30) \)

• Step 1: Check \( x + y\): \( 40+30 = 70 \), and \( 70\geq65 \), so it satisfies \( x + y\geq65 \).
• Step 2: Check \( 20(x + y) \): \( 20\times(40 + 30)=20\times70 = 1400 \), and \( 1400>1000 \), so it satisfies \( 20(x + y)>1000 \).
• Step 3: Check \( x\geq0 \) and \( y\geq0 \): \( 40\geq0 \) and \( 30\geq0 \).

Meaning: Jeremy worked 40 hours and his dad worked 30 hours. Their total hours are 70 (which is more than the minimum of 65 hours), and their total earnings are \( 20\times70 = 1400 \) dollars, which is more than $1000.

Answer:

for \( x\geq0 \) and \( y\geq0 \):
The inequalities \( x\geq0 \) and \( y\geq0 \) mean that the number of hours Jeremy worked (\( x \)) and the number of hours his dad worked (\( y \)) cannot be negative, as you can't work a negative number of hours.

Part b: Identify two solutions and explain what they each mean

A solution is a pair of non - negative values \( (x,y) \) that satisfy all the inequalities \( x + y\geq65 \), \( 20(x + y)>1000 \), \( x\geq0 \), and \( y\geq0 \).

Solution 1: \( (x = 30,y = 35) \)

• Step 1: Check \( x + y\): \( 30+35 = 65 \), and \( 65\geq65 \), so it satisfies \( x + y\geq65 \).
• Step 2: Check \( 20(x + y) \): \( 20\times(30 + 35)=20\times65 = 1300 \), and \( 1300>1000 \), so it satisfies \( 20(x + y)>1000 \).
• Step 3: Check \( x\geq0 \) and \( y\geq0 \): \( 30\geq0 \) and \( 35\geq0 \).

Meaning: Jeremy worked 30 hours and his dad worked 35 hours. Their total hours are 65 (which meets the minimum - hour requirement), and their total earnings are \( 20\times65 = 1300 \) dollars, which is more than $1000.

Solution 2: \( (x = 40,y = 30) \)

• Step 1: Check \( x + y\): \( 40+30 = 70 \), and \( 70\geq65 \), so it satisfies \( x + y\geq65 \).
• Step 2: Check \( 20(x + y) \): \( 20\times(40 + 30)=20\times70 = 1400 \), and \( 1400>1000 \), so it satisfies \( 20(x + y)>1000 \).
• Step 3: Check \( x\geq0 \) and \( y\geq0 \): \( 40\geq0 \) and \( 30\geq0 \).

Meaning: Jeremy worked 40 hours and his dad worked 30 hours. Their total hours are 70 (which is more than the minimum of 65 hours), and their total earnings are \( 20\times70 = 1400 \) dollars, which is more than $1000.