Solved on Feb 10, 2024

Find $q$ when $f=108$ and $h=6$ , given $f$ varies jointly as $q^2$ and $h$ , and $f=64$ when $q=4$ and $h=2$ .

STEP 1

Assumptions
1. The function $f$ varies jointly as $q^2$ and $h$ .
2. $f = 64$ when $q = 4$ and $h = 2$ .
3. We need to find the value of $q$ when $f = 108$ and $h = 6$ .

STEP 2

Since $f$ varies jointly as $q^2$ and $h$ , we can write this relationship as:
$f = kq^2h$
where $k$ is the constant of proportionality.

STEP 3

Use the given values of $f$ , $q$ , and $h$ to find the constant $k$ .
$64 = k \cdot 4^2 \cdot 2$

STEP 4

Calculate the value of $k$ .
$64 = k \cdot 16 \cdot 2$
$64 = 32k$
$k = \frac{64}{32}$
$k = 2$

STEP 5

Now that we have the value of $k$ , we can use it to find $q$ when $f = 108$ and $h = 6$ .
$108 = 2q^2 \cdot 6$

STEP 6

Simplify the equation to solve for $q^2$ .
$108 = 12q^2$
$\frac{108}{12} = q^2$
$9 = q^2$

STEP 7

Take the square root of both sides to solve for $q$ .
$q = \sqrt{9}$
$q = 3$
The value of $q$ when $f = 108$ and $h = 6$ is $3$ .

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Find $q$ when $f=108$ and $h=6$ , given $f$ varies jointly as $q^2$ and $h$ , and $f=64$ when $q=4$ and $h=2$ .

STEP 1

Assumptions
1. The function $f$ varies jointly as $q^2$ and $h$ .
2. $f = 64$ when $q = 4$ and $h = 2$ .
3. We need to find the value of $q$ when $f = 108$ and $h = 6$ .

STEP 2

Since $f$ varies jointly as $q^2$ and $h$ , we can write this relationship as:
$f = kq^2h$
where $k$ is the constant of proportionality.

STEP 3

Use the given values of $f$ , $q$ , and $h$ to find the constant $k$ .
$64 = k \cdot 4^2 \cdot 2$

STEP 4

Calculate the value of $k$ .
$64 = k \cdot 16 \cdot 2$
$64 = 32k$
$k = \frac{64}{32}$
$k = 2$

STEP 5

Now that we have the value of $k$ , we can use it to find $q$ when $f = 108$ and $h = 6$ .
$108 = 2q^2 \cdot 6$

STEP 6

Simplify the equation to solve for $q^2$ .
$108 = 12q^2$
$\frac{108}{12} = q^2$
$9 = q^2$

STEP 7

Take the square root of both sides to solve for $q$ .
$q = \sqrt{9}$
$q = 3$
The value of $q$ when $f = 108$ and $h = 6$ is $3$ .

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Find qqq when f=108f=108f=108 and h=6h=6h=6, given fff varies jointly as q2q^2q2 and hhh, and f=64f=64f=64 when q=4q=4q=4 and h=2h=2h=2.

STEP 1

Assumptions1. The function fff varies jointly as q2q^2q2 and hhh.2. f=64f = 64f=64 when q=4q = 4q=4 and h=2h = 2h=2.3. We need to find the value of qqq when f=108f = 108f=108 and h=6h = 6h=6.

STEP 2

Since fff varies jointly as q2q^2q2 and hhh, we can write this relationship as:f=kq2hf = kq^2hf=kq2hwhere kkk is the constant of proportionality.

STEP 3

Use the given values of fff, qqq, and hhh to find the constant kkk.64=k⋅42⋅264 = k \cdot 4^2 \cdot 264=k⋅42⋅2

STEP 4

Calculate the value of kkk.64=k⋅16⋅264 = k \cdot 16 \cdot 264=k⋅16⋅264=32k64 = 32k64=32kk=6432k = \frac{64}{32}k=3264​k=2k = 2k=2

STEP 5

Now that we have the value of kkk, we can use it to find qqq when f=108f = 108f=108 and h=6h = 6h=6.108=2q2⋅6108 = 2q^2 \cdot 6108=2q2⋅6

STEP 6

Simplify the equation to solve for q2q^2q2.108=12q2108 = 12q^2108=12q210812=q2\frac{108}{12} = q^212108​=q29=q29 = q^29=q2

STEP 7

Take the square root of both sides to solve for qqq.q=9q = \sqrt{9}q=9​q=3q = 3q=3The value of qqq when f=108f = 108f=108 and h=6h = 6h=6 is 333.

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Find $q$ when $f=108$ and $h=6$ , given $f$ varies jointly as $q^2$ and $h$ , and $f=64$ when $q=4$ and $h=2$ .

Assumptions
1. The function $f$ varies jointly as $q^2$ and $h$ .
2. $f = 64$ when $q = 4$ and $h = 2$ .
3. We need to find the value of $q$ when $f = 108$ and $h = 6$ .

Since $f$ varies jointly as $q^2$ and $h$ , we can write this relationship as:
$f = kq^2h$
where $k$ is the constant of proportionality.

Use the given values of $f$ , $q$ , and $h$ to find the constant $k$ .
$64 = k \cdot 4^2 \cdot 2$

Calculate the value of $k$ .
$64 = k \cdot 16 \cdot 2$
$64 = 32k$
$k = \frac{64}{32}$
$k = 2$

Now that we have the value of $k$ , we can use it to find $q$ when $f = 108$ and $h = 6$ .
$108 = 2q^2 \cdot 6$

Simplify the equation to solve for $q^2$ .
$108 = 12q^2$
$\frac{108}{12} = q^2$
$9 = q^2$

Take the square root of both sides to solve for $q$ .
$q = \sqrt{9}$
$q = 3$
The value of $q$ when $f = 108$ and $h = 6$ is $3$ .