Math / Algebra

Question23. Question from 1.12: Modeling Variation
The cost $C$ of printing a magazine is jointly proportional to the number of pages $p$ in the magazine and the number of magazines printed $m$ . (a) Write an equation that expresses this joint variation. (Use $k$ for the constant of proportionality.) $\square$ (b) Find the constant of proportionality if the printing cost is $\$ 55,037.5$ for 3700 copies of a 119 -page magazine. $k=$ $\square$ (c) How much would the printing cost be for 5500 copies of a 52 -page magazine? \\square$

Studdy Solution

STEP 1

What is this asking? We need to figure out the relationship between the cost of printing a magazine, the number of pages, and the number of copies printed, and then use this relationship to calculate some printing costs. Watch out! Make sure to correctly set up the equation for joint variation and carefully plug in the given values to find the constant of proportionality.

STEP 2

1. Express the joint variation.
2. Find the constant of proportionality.
3. Calculate the new printing cost.

STEP 3

Alright, so the problem says the cost $C$ is jointly proportional to the number of pages $p$ and the number of magazines printed $m$ .
This tells us that $C$ changes proportionally with both $p$ and $m$ .

STEP 4

That means if we double the number of pages, the cost doubles.
If we triple the number of magazines, the cost triples!
If we do both, the cost gets multiplied by six!

STEP 5

We can write this relationship as an equation: $C = k \cdot p \cdot m$ , where $k$ is our constant of proportionality.
This constant tells us how much the cost changes for each page and each magazine printed.

STEP 6

We're given that the printing cost is $\$55,037.5$ for 3700 copies of a 119-page magazine.
Let's plug these values into our equation: $55037.5 = k \cdot 119 \cdot 3700$ .

STEP 7

To find $k$ , we need to isolate it.
Let's multiply $119$ and $3700$ to get $440300$ .
So, our equation becomes $55037.5 = k \cdot 440300$ .

STEP 8

Now, we can divide both sides of the equation by $440300$ to find $k$ : $k = \frac{55037.5}{440300} = 0.125$ So, our constant of proportionality, $k$ , is 0.125!
This means it costs $\$0.125$ to print one page of one magazine.

STEP 9

Now we want to know the printing cost for 5500 copies of a 52-page magazine.
We can use our equation $C = k \cdot p \cdot m$ with our newfound $k$ value!

STEP 10

Plugging in $k = 0.125$ , $p = 52$ , and $m = 5500$ , we get: $C = 0.125 \cdot 52 \cdot 5500$

STEP 11

Let's calculate this: $C = 0.125 \cdot 52 \cdot 5500 = 6.5 \cdot 5500 = 35750$ So, the printing cost for 5500 copies of a 52-page magazine is $\$35750$ !

STEP 12

(a) $C = k \cdot p \cdot m$ (b) $k = 0.125$ (c) $\$35750$

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Studdy Solution

STEP 1

STEP 2

1. Express the joint variation.
2. Find the constant of proportionality.
3. Calculate the new printing cost.

STEP 3

Alright, so the problem says the cost $C$ is jointly proportional to the number of pages $p$ and the number of magazines printed $m$ .
This tells us that $C$ changes proportionally with both $p$ and $m$ .

STEP 4

That means if we double the number of pages, the cost doubles.
If we triple the number of magazines, the cost triples!
If we do both, the cost gets multiplied by six!

STEP 5

We can write this relationship as an equation: $C = k \cdot p \cdot m$ , where $k$ is our constant of proportionality.
This constant tells us how much the cost changes for each page and each magazine printed.

STEP 6

We're given that the printing cost is $\$55,037.5$ for 3700 copies of a 119-page magazine.
Let's plug these values into our equation: $55037.5 = k \cdot 119 \cdot 3700$ .

STEP 7

To find $k$ , we need to isolate it.
Let's multiply $119$ and $3700$ to get $440300$ .
So, our equation becomes $55037.5 = k \cdot 440300$ .

STEP 8

Now, we can divide both sides of the equation by $440300$ to find $k$ : $k = \frac{55037.5}{440300} = 0.125$ So, our constant of proportionality, $k$ , is 0.125!
This means it costs $\$0.125$ to print one page of one magazine.

STEP 9

Now we want to know the printing cost for 5500 copies of a 52-page magazine.
We can use our equation $C = k \cdot p \cdot m$ with our newfound $k$ value!

STEP 10

Plugging in $k = 0.125$ , $p = 52$ , and $m = 5500$ , we get: $C = 0.125 \cdot 52 \cdot 5500$

STEP 11

Let's calculate this: $C = 0.125 \cdot 52 \cdot 5500 = 6.5 \cdot 5500 = 35750$ So, the printing cost for 5500 copies of a 52-page magazine is $\$35750$ !

STEP 12

(a) $C = k \cdot p \cdot m$ (b) $k = 0.125$ (c) $\$35750$

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Studdy Solution

STEP 1

STEP 2

1. Express the joint variation.2. Find the constant of proportionality.3. Calculate the new printing cost.

STEP 3

Alright, so the problem says the cost CCC is *jointly proportional* to the number of pages ppp and the number of magazines printed mmm.This tells us that CCC changes proportionally with *both* ppp and mmm.

STEP 4

That means if we double the number of pages, the cost doubles.If we triple the number of magazines, the cost triples!If we do *both*, the cost gets multiplied by six!

STEP 5

We can write this relationship as an equation: C=k⋅p⋅mC = k \cdot p \cdot mC=k⋅p⋅m, where kkk is our **constant of proportionality**.This constant tells us how much the cost changes for each page and each magazine printed.

STEP 6

We're given that the printing cost is $55,037.5\$55,037.5$55,037.5 for **3700** copies of a **119**-page magazine.Let's plug these values into our equation: 55037.5=k⋅119⋅370055037.5 = k \cdot 119 \cdot 370055037.5=k⋅119⋅3700.

STEP 7

To find kkk, we need to isolate it.Let's multiply 119119119 and 370037003700 to get 440300440300440300.So, our equation becomes 55037.5=k⋅44030055037.5 = k \cdot 44030055037.5=k⋅440300.

STEP 8

Now, we can divide both sides of the equation by 440300440300440300 to find kkk: k=55037.5440300=0.125k = \frac{55037.5}{440300} = 0.125k=44030055037.5​=0.125 So, our **constant of proportionality**, kkk, is **0.125**!This means it costs $0.125\$0.125$0.125 to print one page of one magazine.

STEP 9

Now we want to know the printing cost for **5500** copies of a **52**-page magazine.We can use our equation C=k⋅p⋅mC = k \cdot p \cdot mC=k⋅p⋅m with our newfound kkk value!

STEP 10

Plugging in k=0.125k = 0.125k=0.125, p=52p = 52p=52, and m=5500m = 5500m=5500, we get: C=0.125⋅52⋅5500C = 0.125 \cdot 52 \cdot 5500C=0.125⋅52⋅5500

STEP 11

Let's calculate this: C=0.125⋅52⋅5500=6.5⋅5500=35750C = 0.125 \cdot 52 \cdot 5500 = 6.5 \cdot 5500 = 35750C=0.125⋅52⋅5500=6.5⋅5500=35750 So, the printing cost for 5500 copies of a 52-page magazine is $35750\$35750$35750!

STEP 12

(a) C=k⋅p⋅mC = k \cdot p \cdot mC=k⋅p⋅m (b) k=0.125k = 0.125k=0.125 (c) $35750\$35750$35750

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1. Express the joint variation.
2. Find the constant of proportionality.
3. Calculate the new printing cost.

Alright, so the problem says the cost $C$ is jointly proportional to the number of pages $p$ and the number of magazines printed $m$ .
This tells us that $C$ changes proportionally with both $p$ and $m$ .

That means if we double the number of pages, the cost doubles.
If we triple the number of magazines, the cost triples!
If we do both, the cost gets multiplied by six!

We can write this relationship as an equation: $C = k \cdot p \cdot m$ , where $k$ is our constant of proportionality.
This constant tells us how much the cost changes for each page and each magazine printed.

We're given that the printing cost is $\$55,037.5$ for 3700 copies of a 119-page magazine.
Let's plug these values into our equation: $55037.5 = k \cdot 119 \cdot 3700$ .

To find $k$ , we need to isolate it.
Let's multiply $119$ and $3700$ to get $440300$ .
So, our equation becomes $55037.5 = k \cdot 440300$ .

Now, we can divide both sides of the equation by $440300$ to find $k$ : $k = \frac{55037.5}{440300} = 0.125$ So, our constant of proportionality, $k$ , is 0.125!
This means it costs $\$0.125$ to print one page of one magazine.

Now we want to know the printing cost for 5500 copies of a 52-page magazine.
We can use our equation $C = k \cdot p \cdot m$ with our newfound $k$ value!

Plugging in $k = 0.125$ , $p = 52$ , and $m = 5500$ , we get: $C = 0.125 \cdot 52 \cdot 5500$

Let's calculate this: $C = 0.125 \cdot 52 \cdot 5500 = 6.5 \cdot 5500 = 35750$ So, the printing cost for 5500 copies of a 52-page magazine is $\$35750$ !

(a) $C = k \cdot p \cdot m$ (b) $k = 0.125$ (c) $\$35750$