Solved on Jan 23, 2024

Find the formula for $C$ as a function of $G$ if $C$ is proportional to $G^2$ and $C=40$ when $G=5$ . Then use the formula to find $G$ when $C=10$ .
$\mathbf{C} = kG^2$ $k = 40/5^2 = 1.6$ $G = \sqrt{10/1.6} = 2.5$

STEP 1

Assumptions
1. $C$ is proportional to the square of $G$ .
2. $C = 40$ when $G = 5$ .
3. We need to find the constant of proportionality $k$ .
4. We need to write $C$ as a function of $G$ .
5. We need to find $G$ when $C = 10$ .

STEP 2

Since $C$ is proportional to the square of $G$ , we can write the relationship as:
$C = kG^2$
where $k$ is the constant of proportionality.

STEP 3

To find the value of $k$ , we use the given values of $C$ and $G$ when $C = 40$ and $G = 5$ .
$40 = k \cdot 5^2$

STEP 4

Solve for $k$ by dividing both sides of the equation by $5^2$ .
$k = \frac{40}{5^2}$

STEP 5

Calculate the value of $k$ .
$k = \frac{40}{25}$

STEP 6

Simplify the fraction to find the value of $k$ .
$k = \frac{8}{5}$

STEP 7

Now that we have the value of $k$ , we can write the formula for $C$ as a function of $G$ .
$C = \frac{8}{5}G^2$

STEP 8

To find $G$ when $C = 10$ , we substitute $10$ for $C$ in the formula.
$10 = \frac{8}{5}G^2$

STEP 9

Multiply both sides of the equation by $\frac{5}{8}$ to solve for $G^2$ .
$G^2 = \frac{10 \cdot 5}{8}$

STEP 10

Calculate the value of $G^2$ .
$G^2 = \frac{50}{8}$

STEP 11

Simplify the fraction to find $G^2$ .
$G^2 = \frac{25}{4}$

STEP 12

Take the square root of both sides to solve for $G$ .
$G = \sqrt{\frac{25}{4}}$

STEP 13

Calculate the value of $G$ .
$G = \frac{5}{2}$
The GENERAL formula before solving for $k$ is: $\mathbf{C} = kG^2$
The value for $k$ is: $k = \frac{8}{5}$
Using the formula to find $G$ when $C = 10$ : $\mathrm{G} = \frac{5}{2} \quad \text{and} \quad \mathbf{G} = 2.5$

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Find the formula for $C$ as a function of $G$ if $C$ is proportional to $G^2$ and $C=40$ when $G=5$ . Then use the formula to find $G$ when $C=10$ .
$\mathbf{C} = kG^2$ $k = 40/5^2 = 1.6$ $G = \sqrt{10/1.6} = 2.5$

STEP 1

Assumptions
1. $C$ is proportional to the square of $G$ .
2. $C = 40$ when $G = 5$ .
3. We need to find the constant of proportionality $k$ .
4. We need to write $C$ as a function of $G$ .
5. We need to find $G$ when $C = 10$ .

STEP 2

Since $C$ is proportional to the square of $G$ , we can write the relationship as:
$C = kG^2$
where $k$ is the constant of proportionality.

STEP 3

To find the value of $k$ , we use the given values of $C$ and $G$ when $C = 40$ and $G = 5$ .
$40 = k \cdot 5^2$

STEP 4

Solve for $k$ by dividing both sides of the equation by $5^2$ .
$k = \frac{40}{5^2}$

STEP 5

Calculate the value of $k$ .
$k = \frac{40}{25}$

STEP 6

Simplify the fraction to find the value of $k$ .
$k = \frac{8}{5}$

STEP 7

Now that we have the value of $k$ , we can write the formula for $C$ as a function of $G$ .
$C = \frac{8}{5}G^2$

STEP 8

To find $G$ when $C = 10$ , we substitute $10$ for $C$ in the formula.
$10 = \frac{8}{5}G^2$

STEP 9

Multiply both sides of the equation by $\frac{5}{8}$ to solve for $G^2$ .
$G^2 = \frac{10 \cdot 5}{8}$

STEP 10

Calculate the value of $G^2$ .
$G^2 = \frac{50}{8}$

STEP 11

Simplify the fraction to find $G^2$ .
$G^2 = \frac{25}{4}$

STEP 12

Take the square root of both sides to solve for $G$ .
$G = \sqrt{\frac{25}{4}}$

STEP 13

Calculate the value of $G$ .
$G = \frac{5}{2}$
The GENERAL formula before solving for $k$ is: $\mathbf{C} = kG^2$
The value for $k$ is: $k = \frac{8}{5}$
Using the formula to find $G$ when $C = 10$ : $\mathrm{G} = \frac{5}{2} \quad \text{and} \quad \mathbf{G} = 2.5$

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Find the formula for CCC as a function of GGG if CCC is proportional to G2G^2G2 and C=40C=40C=40 when G=5G=5G=5. Then use the formula to find GGG when C=10C=10C=10.C=kG2\mathbf{C} = kG^2C=kG2 k=40/52=1.6k = 40/5^2 = 1.6k=40/52=1.6 G=10/1.6=2.5G = \sqrt{10/1.6} = 2.5G=10/1.6​=2.5

STEP 1

Assumptions1. C C C is proportional to the square of G G G.2. C=40 C = 40 C=40 when G=5 G = 5 G=5.3. We need to find the constant of proportionality k k k.4. We need to write C C C as a function of G G G.5. We need to find G G G when C=10 C = 10 C=10.

STEP 2

Since C C C is proportional to the square of G G G, we can write the relationship as:C=kG2 C = kG^2 C=kG2where k k k is the constant of proportionality.

STEP 3

To find the value of k k k, we use the given values of C C C and G G G when C=40 C = 40 C=40 and G=5 G = 5 G=5.40=k⋅52 40 = k \cdot 5^2 40=k⋅52

STEP 4

Solve for k k k by dividing both sides of the equation by 52 5^2 52.k=4052 k = \frac{40}{5^2} k=5240​

STEP 5

Calculate the value of k k k.k=4025 k = \frac{40}{25} k=2540​

STEP 6

Simplify the fraction to find the value of k k k.k=85 k = \frac{8}{5} k=58​

STEP 7

Now that we have the value of k k k, we can write the formula for C C C as a function of G G G.C=85G2 C = \frac{8}{5}G^2 C=58​G2

STEP 8

To find G G G when C=10 C = 10 C=10, we substitute 10 10 10 for C C C in the formula.10=85G2 10 = \frac{8}{5}G^2 10=58​G2

STEP 9

Multiply both sides of the equation by 58 \frac{5}{8} 85​ to solve for G2 G^2 G2.G2=10⋅58 G^2 = \frac{10 \cdot 5}{8} G2=810⋅5​

STEP 10

Calculate the value of G2 G^2 G2.G2=508 G^2 = \frac{50}{8} G2=850​

STEP 11

Simplify the fraction to find G2 G^2 G2.G2=254 G^2 = \frac{25}{4} G2=425​

STEP 12

Take the square root of both sides to solve for G G G.G=254 G = \sqrt{\frac{25}{4}} G=425​​

STEP 13

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Find the formula for $C$ as a function of $G$ if $C$ is proportional to $G^2$ and $C=40$ when $G=5$ . Then use the formula to find $G$ when $C=10$ .
$\mathbf{C} = kG^2$ $k = 40/5^2 = 1.6$ $G = \sqrt{10/1.6} = 2.5$

Assumptions
1. $C$ is proportional to the square of $G$ .
2. $C = 40$ when $G = 5$ .
3. We need to find the constant of proportionality $k$ .
4. We need to write $C$ as a function of $G$ .
5. We need to find $G$ when $C = 10$ .

Since $C$ is proportional to the square of $G$ , we can write the relationship as:
$C = kG^2$
where $k$ is the constant of proportionality.

To find the value of $k$ , we use the given values of $C$ and $G$ when $C = 40$ and $G = 5$ .
$40 = k \cdot 5^2$

Solve for $k$ by dividing both sides of the equation by $5^2$ .
$k = \frac{40}{5^2}$

Calculate the value of $k$ .
$k = \frac{40}{25}$

Simplify the fraction to find the value of $k$ .
$k = \frac{8}{5}$

Now that we have the value of $k$ , we can write the formula for $C$ as a function of $G$ .
$C = \frac{8}{5}G^2$

To find $G$ when $C = 10$ , we substitute $10$ for $C$ in the formula.
$10 = \frac{8}{5}G^2$

Multiply both sides of the equation by $\frac{5}{8}$ to solve for $G^2$ .
$G^2 = \frac{10 \cdot 5}{8}$

Calculate the value of $G^2$ .
$G^2 = \frac{50}{8}$

Simplify the fraction to find $G^2$ .
$G^2 = \frac{25}{4}$

Take the square root of both sides to solve for $G$ .
$G = \sqrt{\frac{25}{4}}$