Solved on Jan 20, 2024

Find the change in $x$ and $y$ when $y$ varies at a constant rate of 4 with respect to $x$ , for the given ranges of $x$ .
a. If $x$ varies from $x=3.7$ to $x=8.8$ , then: i. The change in $x$ is $\Delta x=5.1$ ii. The corresponding change in $y$ is $\Delta y=20.4$
b. If $x$ varies from $x=2$ to $x=-7$ , then: i. The change in $x$ is $\Delta x=-9$ ii. The corresponding change in $y$ is $\Delta y=-36$

STEP 1

Assumptions
1. $y$ varies at a constant rate of change of 4 with respect to $x$ .
2. This means that for every unit increase in $x$ , $y$ increases by 4 units.
3. The change in $x$ is denoted by $\Delta x$ and the change in $y$ is denoted by $\Delta y$ .

STEP 2

To find the change in $x$ when $x$ varies from $x=3.7$ to $x=8.8$ , we subtract the initial value of $x$ from the final value of $x$ .
$\Delta x = x_{final} - x_{initial}$

STEP 3

Now, plug in the given values for $x_{initial}$ and $x_{final}$ to calculate $\Delta x$ .
$\Delta x = 8.8 - 3.7$

STEP 4

Calculate the change in $x$ .
$\Delta x = 8.8 - 3.7 = 5.1$

STEP 5

To find the corresponding change in $y$ , we multiply the change in $x$ by the constant rate of change of $y$ with respect to $x$ .
$\Delta y = \Delta x \times \text{Rate of change of } y$

STEP 6

Now, plug in the values for $\Delta x$ and the rate of change of $y$ to calculate $\Delta y$ .
$\Delta y = 5.1 \times 4$

STEP 7

Calculate the corresponding change in $y$ .
$\Delta y = 5.1 \times 4 = 20.4$

STEP 8

To find the change in $x$ when $x$ varies from $x=2$ to $x=-7$ , we again subtract the initial value of $x$ from the final value of $x$ .
$\Delta x = x_{final} - x_{initial}$

STEP 9

Now, plug in the given values for $x_{initial}$ and $x_{final}$ to calculate $\Delta x$ .
$\Delta x = -7 - 2$

STEP 10

Calculate the change in $x$ .
$\Delta x = -7 - 2 = -9$

STEP 11

To find the corresponding change in $y$ when $x$ changes by $\Delta x = -9$ , we multiply the change in $x$ by the constant rate of change of $y$ with respect to $x$ .
$\Delta y = \Delta x \times \text{Rate of change of } y$

STEP 12

Now, plug in the values for $\Delta x$ and the rate of change of $y$ to calculate $\Delta y$ .
$\Delta y = -9 \times 4$

STEP 13

Calculate the corresponding change in $y$ .
$\Delta y = -9 \times 4 = -36$
The solutions are: a. i. $\Delta x = 5.1$ ii. $\Delta y = 20.4$ b. i. $\Delta x = -9$ ii. $\Delta y = -36$

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STEP 1

Assumptions
1. $y$ varies at a constant rate of change of 4 with respect to $x$ .
2. This means that for every unit increase in $x$ , $y$ increases by 4 units.
3. The change in $x$ is denoted by $\Delta x$ and the change in $y$ is denoted by $\Delta y$ .

STEP 2

To find the change in $x$ when $x$ varies from $x=3.7$ to $x=8.8$ , we subtract the initial value of $x$ from the final value of $x$ .
$\Delta x = x_{final} - x_{initial}$

STEP 3

Now, plug in the given values for $x_{initial}$ and $x_{final}$ to calculate $\Delta x$ .
$\Delta x = 8.8 - 3.7$

STEP 4

Calculate the change in $x$ .
$\Delta x = 8.8 - 3.7 = 5.1$

STEP 5

To find the corresponding change in $y$ , we multiply the change in $x$ by the constant rate of change of $y$ with respect to $x$ .
$\Delta y = \Delta x \times \text{Rate of change of } y$

STEP 6

Now, plug in the values for $\Delta x$ and the rate of change of $y$ to calculate $\Delta y$ .
$\Delta y = 5.1 \times 4$

STEP 7

Calculate the corresponding change in $y$ .
$\Delta y = 5.1 \times 4 = 20.4$

STEP 8

To find the change in $x$ when $x$ varies from $x=2$ to $x=-7$ , we again subtract the initial value of $x$ from the final value of $x$ .
$\Delta x = x_{final} - x_{initial}$

STEP 9

Now, plug in the given values for $x_{initial}$ and $x_{final}$ to calculate $\Delta x$ .
$\Delta x = -7 - 2$

STEP 10

Calculate the change in $x$ .
$\Delta x = -7 - 2 = -9$

STEP 11

To find the corresponding change in $y$ when $x$ changes by $\Delta x = -9$ , we multiply the change in $x$ by the constant rate of change of $y$ with respect to $x$ .
$\Delta y = \Delta x \times \text{Rate of change of } y$

STEP 12

Now, plug in the values for $\Delta x$ and the rate of change of $y$ to calculate $\Delta y$ .
$\Delta y = -9 \times 4$

STEP 13

Calculate the corresponding change in $y$ .
$\Delta y = -9 \times 4 = -36$
The solutions are: a. i. $\Delta x = 5.1$ ii. $\Delta y = 20.4$ b. i. $\Delta x = -9$ ii. $\Delta y = -36$

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STEP 1

Assumptions1. yyy varies at a constant rate of change of 4 with respect to xxx.2. This means that for every unit increase in xxx, yyy increases by 4 units.3. The change in xxx is denoted by Δx\Delta xΔx and the change in yyy is denoted by Δy\Delta yΔy.

STEP 2

To find the change in xxx when xxx varies from x=3.7x=3.7x=3.7 to x=8.8x=8.8x=8.8, we subtract the initial value of xxx from the final value of xxx.Δx=xfinal−xinitial\Delta x = x_{final} - x_{initial}Δx=xfinal​−xinitial​

STEP 3

Now, plug in the given values for xinitialx_{initial}xinitial​ and xfinalx_{final}xfinal​ to calculate Δx\Delta xΔx.Δx=8.8−3.7\Delta x = 8.8 - 3.7Δx=8.8−3.7

STEP 4

Calculate the change in xxx.Δx=8.8−3.7=5.1\Delta x = 8.8 - 3.7 = 5.1Δx=8.8−3.7=5.1

STEP 5

To find the corresponding change in yyy, we multiply the change in xxx by the constant rate of change of yyy with respect to xxx.Δy=Δx×Rate of change of y\Delta y = \Delta x \times \text{Rate of change of } yΔy=Δx×Rate of change of y

STEP 6

Now, plug in the values for Δx\Delta xΔx and the rate of change of yyy to calculate Δy\Delta yΔy.Δy=5.1×4\Delta y = 5.1 \times 4Δy=5.1×4

STEP 7

Calculate the corresponding change in yyy.Δy=5.1×4=20.4\Delta y = 5.1 \times 4 = 20.4Δy=5.1×4=20.4

STEP 8

To find the change in xxx when xxx varies from x=2x=2x=2 to x=−7x=-7x=−7, we again subtract the initial value of xxx from the final value of xxx.Δx=xfinal−xinitial\Delta x = x_{final} - x_{initial}Δx=xfinal​−xinitial​

STEP 9

Now, plug in the given values for xinitialx_{initial}xinitial​ and xfinalx_{final}xfinal​ to calculate Δx\Delta xΔx.Δx=−7−2\Delta x = -7 - 2Δx=−7−2

STEP 10

Calculate the change in xxx.Δx=−7−2=−9\Delta x = -7 - 2 = -9Δx=−7−2=−9

STEP 11

To find the corresponding change in yyy when xxx changes by Δx=−9\Delta x = -9Δx=−9, we multiply the change in xxx by the constant rate of change of yyy with respect to xxx.Δy=Δx×Rate of change of y\Delta y = \Delta x \times \text{Rate of change of } yΔy=Δx×Rate of change of y

STEP 12

Now, plug in the values for Δx\Delta xΔx and the rate of change of yyy to calculate Δy\Delta yΔy.Δy=−9×4\Delta y = -9 \times 4Δy=−9×4

STEP 13

Calculate the corresponding change in yyy.Δy=−9×4=−36\Delta y = -9 \times 4 = -36Δy=−9×4=−36The solutions are: a. i. Δx=5.1\Delta x = 5.1Δx=5.1 ii. Δy=20.4\Delta y = 20.4Δy=20.4 b. i. Δx=−9\Delta x = -9Δx=−9 ii. Δy=−36\Delta y = -36Δy=−36

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Assumptions
1. $y$ varies at a constant rate of change of 4 with respect to $x$ .
2. This means that for every unit increase in $x$ , $y$ increases by 4 units.
3. The change in $x$ is denoted by $\Delta x$ and the change in $y$ is denoted by $\Delta y$ .

To find the change in $x$ when $x$ varies from $x=3.7$ to $x=8.8$ , we subtract the initial value of $x$ from the final value of $x$ .
$\Delta x = x_{final} - x_{initial}$

Now, plug in the given values for $x_{initial}$ and $x_{final}$ to calculate $\Delta x$ .
$\Delta x = 8.8 - 3.7$

Calculate the change in $x$ .
$\Delta x = 8.8 - 3.7 = 5.1$

To find the corresponding change in $y$ , we multiply the change in $x$ by the constant rate of change of $y$ with respect to $x$ .
$\Delta y = \Delta x \times \text{Rate of change of } y$

Now, plug in the values for $\Delta x$ and the rate of change of $y$ to calculate $\Delta y$ .
$\Delta y = 5.1 \times 4$

Calculate the corresponding change in $y$ .
$\Delta y = 5.1 \times 4 = 20.4$

To find the change in $x$ when $x$ varies from $x=2$ to $x=-7$ , we again subtract the initial value of $x$ from the final value of $x$ .
$\Delta x = x_{final} - x_{initial}$

Now, plug in the given values for $x_{initial}$ and $x_{final}$ to calculate $\Delta x$ .
$\Delta x = -7 - 2$

Calculate the change in $x$ .
$\Delta x = -7 - 2 = -9$

To find the corresponding change in $y$ when $x$ changes by $\Delta x = -9$ , we multiply the change in $x$ by the constant rate of change of $y$ with respect to $x$ .
$\Delta y = \Delta x \times \text{Rate of change of } y$

Now, plug in the values for $\Delta x$ and the rate of change of $y$ to calculate $\Delta y$ .
$\Delta y = -9 \times 4$

Calculate the corresponding change in $y$ .
$\Delta y = -9 \times 4 = -36$
The solutions are: a. i. $\Delta x = 5.1$ ii. $\Delta y = 20.4$ b. i. $\Delta x = -9$ ii. $\Delta y = -36$