4. the table shows the number of hours worked cutting grass to the amount paid.\n| number of hours | 1 | 3 |…

4. the table shows the number of hours worked cutting grass to the amount paid.\n| number of hours | 1 | 3 | 2 | 4 | 2 |\n| amount paid | 10 | 45 | 16 | 68 | 20 |\na) enter the data in desmos to determine the linear equation of best fit.\nb) use the equation to predict the amount paid for someone who works 6 hours.\nc) use the equation to predict the number of hours worked for someone who get paid $72.

4. the table shows the number of hours worked cutting grass to the amount paid.\n| number of hours | 1 | 3 | 2 | 4 | 2 |\n| amount paid | 10 | 45 | 16 | 68 | 20 |\na) enter the data in desmos to determine the linear equation of best fit.\nb) use the equation to predict the amount paid for someone who works 6 hours.\nc) use the equation to predict the number of hours worked for someone who get paid $72.

Answer

Explanation:

Step 1: Define variables and calculate sums

Let ( x ) = hours worked, ( y ) = amount paid.
Data points: ((1,10), (3,45), (2,16), (4,68), (2,20)).
Calculate ( \sum x = 1+3+2+4+2=12 ), ( \sum y = 10+45+16+68+20=159 ),
( \sum xy = 1(10)+3(45)+2(16)+4(68)+2(20)=10+135+32+272+40=489 ),
( \sum x^2 = 1^2+3^2+2^2+4^2+2^2=1+9+4+16+4=34 ), ( n=5 ).

Step 2: Calculate slope ( b )

Use formula ( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} ):
[ b = \frac{5(489) - 12(159)}{5(34) - 12^2} = \frac{2445 - 1908}{170 - 144} = \frac{537}{26} \approx 20.6538 ]

Step 3: Calculate intercept ( a )

Use ( a = \bar{y} - b\bar{x} ), where ( \bar{x} = \frac{12}{5}=2.4 ), ( \bar{y} = \frac{159}{5}=31.8 ):
[ a = 31.8 - 20.6538(2.4) \approx 31.8 - 49.5691 = -17.7691 ]

Step 4: Form linear equation

Best fit line: ( y = 20.6538x - 17.7691 ) (rounded for Desmos, exact values may vary slightly depending on tool precision).

Step 5: Predict for 6 hours (part b)

Substitute ( x=6 ):
[ y = 20.6538(6) - 17.7691 \approx 123.9228 - 17.7691 \approx 106.1537 \approx $106.15 ]

Step 6: Predict hours for $72 (part c)

Set ( y=72 ) and solve for ( x ):
[ 72 = 20.6538x - 17.7691 ]
[ 20.6538x = 72 + 17.7691 = 89.7691 ]
[ x \approx \frac{89.7691}{20.6538} \approx 4.346 \approx 4.35 \text{ hours} ]

Answer:

a) Linear equation of best fit (via Desmos): ( y = 20.65x - 17.77 ) (approximate, exact values depend on tool precision)
b) Predicted amount for 6 hours: $106.15
c) Predicted hours for $72: 4.35 hours