Derek owns a landscape business. He charges a fixed fee of $30 plus $1 per 1,000 square feet of lawn mowed. Derek's earnings (in
dollars) in the past five weeks are {204, 344, 450, 482, 504}. To find the corresponding square footage of lawn mowed, construct a function that models the total area of lawn that Derek mows based on his earnings. Based on Derek's earnings, the corresponding square footage of the lawns he mowed in the past five weeks is
2 answers:
Let
x = earnings, dollars
y = area mowed, in 1000's of square feet.
Then the required function is
y = x - 30
Construct a table that shows Derek's weekly earnings versus area mowed.
week x y
------- -------- ----------
1 204 174
2 344 314
3 450 420
4 482 452
5 504 474
----- ------- -----------
Total 1984 1834
A graph of y versus x is shown below.
From the table, obtain the total area mowed in the past five weeks as the sum of the y columns, which is 1,834 thousand square feet.
Answer: 1,834 thousand square feet.
x = earnings, dollars
y = area mowed, in 1000's of square feet.
Then the required function is
y = x - 30
Construct a table that shows Derek's weekly earnings versus area mowed.
week x y
------- -------- ----------
1 204 174
2 344 314
3 450 420
4 482 452
5 504 474
----- ------- -----------
Total 1984 1834
A graph of y versus x is shown below.
From the table, obtain the total area mowed in the past five weeks as the sum of the y columns, which is 1,834 thousand square feet.
Answer: 1,834 thousand square feet.
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I made my own graph using the given points. Pls. see attachment.
<span>Part A: Using the graph above, create a system of inequalities that only contains points D and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.
I used paint program to graph the points based on the given data.
I used a ruler as guide as I draw my line containing both points D and F and extended the line outward.
The yellow lines above represents the shaded portion of the graph.
y = mx + b
b is the y-intercept. It is the value of y when x is 0. In this case, b = 3.5
m is the slope. It is the change in y divided by the change in x.
point D (2,4) ; point F (-2,3)
change in y: 4 - 3 = 1
change in x: 2 - (-2) = 4
m = 1/4
y = 1/4 x + 3.5
Since we are looking for a system of inequality and the portion shaded is in the upper part of the line, the value of y will be greater than or equal to.
y </span>≥ 1/4x + 3.5<span>
Part B: Explain how to verify that the points D and F are solutions to the system of inequalities created in Part A.
To check the solution given above. We will use the x values of points D and F and check whether its y values are equal to or greater than the solution.</span>
y ≥ 1/4x + 3.5
Point D (2,4) : x = 2 : y ≥ 1/4 * 2 + 3.5 → 0.5 + 3.5 → 4
Point F (-2.3): x = -2 ; y ≥ 1/4 * -2 + 3.5 → -0.5 + 3.5 → 3
<span>
Part C: Chickens can only be raised in the area defined by y > 2x − 2. Explain how you can identify farms in which chickens can be raised.
y > 2x - 2
We will get the coordinates of each point and see if its x value will generate the desired value of y.
A(2,-3) ; x = 2 : </span><span>y > 2(2) - 2 </span>→ 4 - 2 → 2 . DID NOT PASS. -3 is lesser than 2.<span>
B(-3,-4) ; x = -3 : y > 2(-3) - 2 </span>→ -6 - 2 → -8. PASSED. -4 is greater than -8.<span>
C(-4,2) ; x = -4 : y > 2(-4) - 2 </span>→ -8 - 2 → -10. PASSED. 2 is greater than -10.<span>
D(2,4) ; x = 2 : y > 2(2) - 2 </span>→ 4 - 2 → 2. PASSED. 4 is greater than 2.<span>
E(3,1) ; x = 3 : y > 2(3) - 2 </span>→ 6 - 2 → 4. DID NOT PASS. 1 is lesser than 4.<span>
F(-2,3) ; x = -2 ; y > 2(-2) - 2 </span>→ -4 - 2 → -6. PASSED. 3 is greater than -6.
Among the farms, points B,C, D, and F can raise chickens based on the given system of linear inequality.
<span>Part A: Using the graph above, create a system of inequalities that only contains points D and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.
I used paint program to graph the points based on the given data.
I used a ruler as guide as I draw my line containing both points D and F and extended the line outward.
The yellow lines above represents the shaded portion of the graph.
y = mx + b
b is the y-intercept. It is the value of y when x is 0. In this case, b = 3.5
m is the slope. It is the change in y divided by the change in x.
point D (2,4) ; point F (-2,3)
change in y: 4 - 3 = 1
change in x: 2 - (-2) = 4
m = 1/4
y = 1/4 x + 3.5
Since we are looking for a system of inequality and the portion shaded is in the upper part of the line, the value of y will be greater than or equal to.
y </span>≥ 1/4x + 3.5<span>
Part B: Explain how to verify that the points D and F are solutions to the system of inequalities created in Part A.
To check the solution given above. We will use the x values of points D and F and check whether its y values are equal to or greater than the solution.</span>
y ≥ 1/4x + 3.5
Point D (2,4) : x = 2 : y ≥ 1/4 * 2 + 3.5 → 0.5 + 3.5 → 4
Point F (-2.3): x = -2 ; y ≥ 1/4 * -2 + 3.5 → -0.5 + 3.5 → 3
<span>
Part C: Chickens can only be raised in the area defined by y > 2x − 2. Explain how you can identify farms in which chickens can be raised.
y > 2x - 2
We will get the coordinates of each point and see if its x value will generate the desired value of y.
A(2,-3) ; x = 2 : </span><span>y > 2(2) - 2 </span>→ 4 - 2 → 2 . DID NOT PASS. -3 is lesser than 2.<span>
B(-3,-4) ; x = -3 : y > 2(-3) - 2 </span>→ -6 - 2 → -8. PASSED. -4 is greater than -8.<span>
C(-4,2) ; x = -4 : y > 2(-4) - 2 </span>→ -8 - 2 → -10. PASSED. 2 is greater than -10.<span>
D(2,4) ; x = 2 : y > 2(2) - 2 </span>→ 4 - 2 → 2. PASSED. 4 is greater than 2.<span>
E(3,1) ; x = 3 : y > 2(3) - 2 </span>→ 6 - 2 → 4. DID NOT PASS. 1 is lesser than 4.<span>
F(-2,3) ; x = -2 ; y > 2(-2) - 2 </span>→ -4 - 2 → -6. PASSED. 3 is greater than -6.
Among the farms, points B,C, D, and F can raise chickens based on the given system of linear inequality.