Math  /  Algebra

QuestionCost, Revenue \& Profit For these problems, xx will represent the number of items and yy will represent the money. The fixed costs for a certain item are $120\$ 120 per week. The cost to produce each item is $8\$ 8 per item. Using this information, what is the cost equation? Give your answer in slope-intercept form: y=y= \square
The retailer intends to sell each item for $10/\$ 10 / item. Using this information, what is the revenue equation? Give your answer in slope-intercept form: y=y= \square
If in this week 84 items are made, and all items are sold in the week, what are the total costs to the retailer? Cost =$=\$ \square What is the revenue from selling 84 items? Revenue = \ \squareFinally,whatistheprofitforthisretailer?Profit Finally, what is the profit for this retailer? Profit =\$$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to figure out the equations for cost and revenue based on the number of items, then calculate the total cost, revenue, and profit if 84 items are made and sold. Watch out! Don't mix up cost and revenue!
Cost is what the retailer *pays*, and revenue is what the retailer *receives*.

STEP 2

1. Find the Cost Equation
2. Find the Revenue Equation
3. Calculate Total Costs
4. Calculate Revenue
5. Calculate Profit

STEP 3

Alright, let's **define** our cost equation!
We know there's a **fixed cost** of $120\$120 per week, no matter how many items are made.
This is like our starting point.
Then, for each item made, there's an **additional cost** of $8\$8.

STEP 4

So, our **total cost** (yy) is the **fixed cost** plus the **variable cost** (cost per item times the number of items, xx).
This gives us the equation: y=8x+120 y = 8 \cdot x + 120 This is in **slope-intercept form**, where 88 is the **slope** (how much the cost goes up for each item) and 120120 is the **y-intercept** (the cost when no items are made).

STEP 5

Now for the **revenue equation**!
Revenue is the money earned from selling the items.
The retailer sells each item for $10\$10.

STEP 6

So, the **total revenue** (yy) is simply the price per item times the number of items sold (xx): y=10x y = 10 \cdot x Here, the **slope** is 1010 (how much revenue increases for each item sold), and the **y-intercept** is 00 since no revenue is generated if no items are sold.

STEP 7

We're told that **84 items** are made.
Let's plug this value of xx into our **cost equation**: y=884+120 y = 8 \cdot 84 + 120

STEP 8

y=672+120 y = 672 + 120

STEP 9

y=792 y = 792 So, the **total cost** is $792\$792.

STEP 10

All 84 items are sold.
Let's plug x=84x = 84 into our **revenue equation**: y=1084 y = 10 \cdot 84

STEP 11

y=840 y = 840 The **total revenue** is $840\$840.

STEP 12

**Profit** is the difference between **revenue** and **cost**.

STEP 13

Profit=RevenueCost \text{Profit} = \text{Revenue} - \text{Cost} Profit=840792 \text{Profit} = 840 - 792

STEP 14

Profit=48 \text{Profit} = 48 The **profit** is $48\$48.

STEP 15

Cost equation: y=8x+120y = 8x + 120 Revenue equation: y=10xy = 10x Total costs for 84 items: $792\$792 Revenue from selling 84 items: $840\$840 Profit: $48\$48

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