Pen Purchase Cost: Function Of Boxes Bought

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Hey guys! Ever found yourself staring at a math problem and thinking, "How on earth do I turn this into a function?" Well, today we're diving into a super relatable scenario: buying pens! We've got Barbara here, who's making a pen purchase, and we need to figure out the total cost based on how many boxes she grabs. It sounds simple, but it's a fantastic way to wrap our heads around function notation and how it applies to real-world situations. So, let's break down this pen problem and see which mathematical expression truly represents Barbara's total spending. We'll be looking at the cost per box, the discount she snagges, and how to put it all together in a neat little function. By the end of this, you'll be a pro at translating word problems into algebraic expressions, making those pesky math assignments feel like a breeze. We're going to explore how a seemingly simple discount can change the entire way we calculate costs, and how functions are the perfect tool to model these changes. Get ready to flex those math muscles, because we're about to make some serious sense of this pen-tastic problem!

Understanding the Core Problem: Barbara's Pen Shopping Spree

Alright, let's get into the nitty-gritty of Barbara's pen purchase. The initial cost for the first box of pens is $4. This is our starting point, the base price. Now, here's where it gets interesting: for every additional box she buys, she gets a **1discount∗∗.Thismeansthepriceisn′tastraightforwardmultiplicationofthenumberofboxesbyafixedprice.Thepriceperbox∗changes∗dependingonhowmanyboxesshe′salreadybought.Thisisthekeyelementthatmakesitafunctionproblem.Weneedtofindanexpressionthatcapturesthischangingcost.We′relookingforthetotalcost,let′scallit′1 discount**. This means the price isn't a straightforward multiplication of the number of boxes by a fixed price. The price per box *changes* depending on how many boxes she's already bought. This is the key element that makes it a function problem. We need to find an expression that captures this changing cost. We're looking for the total cost, let's call it 'c′,asafunctionofthenumberofboxes,whichwe′lldenoteas′', as a function of the number of boxes, which we'll denote as 'b′.Thisiswherefunctionnotation,like′'. This is where function notation, like 'c = f(b)′,comesintoplay.It′sawaytosaythatthecost′', comes into play. It's a way to say that the cost 'c′∗dependson∗thenumberofboxes′' *depends on* the number of boxes 'b

. So, how do we build this function? Let's think step-by-step. If Barbara buys just one box, the cost is $4. If she buys two boxes, the first box is $4, but the second box gets a $1 discount, so it costs $3. The total would be $4 + $3 = $7. If she buys three boxes, the first is $4, the second is $3, and the third also gets a $1 discount off the original price (or perhaps off the previous box's price? This is where we need to be super careful and read the problem exactly). The problem states, "For every additional box she buys, she gets a $1 discount." This usually implies the discount is applied to the price of that additional box relative to the initial price. So, if the first box is $4, the second is $4 - $1 = $3, and the third is also $4 - $1 = $3. This interpretation seems the most common for these types of problems. Let's stick with that for now. So, for three boxes, the cost would be $4 (first box) + $3 (second box) + $3 (third box) = $10. It's not as simple as just multiplying $4 by the number of boxes, because that wouldn't account for the discount. And it's not as simple as subtracting $1 from the total cost of 4b4b, because the discount applies per additional box, not as a one-time deduction. We need an expression that correctly sums up the cost of each box, considering the discount only kicks in after the very first box.

Deconstructing the Cost: First Box vs. Additional Boxes

Let's really dig into the pricing structure, guys. The first box of pens costs a flat $4. This is our baseline, our entry fee, if you will. Now, the magic happens with the additional boxes. For each box Barbara buys after the first one, she scores a sweet $1 discount. This means the price of every subsequent box is effectively $4 - $1 = 3.Thisiscrucialbecauseittellsuswehavetwodifferentpricepointsatplay:thepriceforthefirstboxandthepriceforalltheotherboxes.Whenwe′rebuildingourfunction′3. This is crucial because it tells us we have two different price points at play: the price for the first box and the price for all the other boxes. When we're building our function 'c = f(b)′,weneedtoaccountforthisdifference.If′', we need to account for this difference. If 'b′representsthetotalnumberofboxes,thenthenumberof∗additional∗boxesis′' represents the total number of boxes, then the number of *additional* boxes is 'b - 1

. This makes sense, right? If she buys 5 boxes, there's 1 first box and 4 additional boxes (5−1=45 - 1 = 4). If she buys only 1 box, there are 0 additional boxes (1−1=01 - 1 = 0). This 'b−1b - 1' term is going to be super important in our calculation. So, the total cost 'cc' will be the cost of the first box PLUS the cost of all the additional boxes. The cost of the first box is always $4. The cost of the additional boxes is the number of additional boxes multiplied by the discounted price per box. That discounted price is 3.So,thecostoftheadditionalboxesis′3. So, the cost of the additional boxes is ' (b - 1) * 3 ′.Puttingitalltogether,thetotalcost′'. Putting it all together, the total cost 'c′wouldbe:′' would be: 'c = 4 + (b - 1) * 3′.Thisexpression∗directlymodels∗thescenariodescribed.Itseparatesthecostofthefirstboxfromthecostoftherest,applyingthediscountonlytothosesubsequentboxes.It′slikesaying,"Okay,payfullpriceforthefirstone,theneverythingelsegetsalittlecheaper."Thisisafundamentalconceptinunderstandinglinearfunctionswheretheremightbeaninitialvalueandaconstantrateofchange(or,inthiscase,aconstantpriceforsubsequentitems).We′renotjustblindlymultiplying′'. This expression *directly models* the scenario described. It separates the cost of the first box from the cost of the rest, applying the discount only to those subsequent boxes. It's like saying, "Okay, pay full price for the first one, then everything else gets a little cheaper." This is a fundamental concept in understanding linear functions where there might be an initial value and a constant rate of change (or, in this case, a constant price for subsequent items). We're not just blindly multiplying 'b by some number; we're carefully constructing the cost based on the conditions given in the problem. It's about translating the English sentence "For every additional box she buys, she gets a $1 discount" into mathematical terms, which is precisely what functions help us do. We're building a model of the real-world transaction.

Evaluating the Options: Finding the Correct Expression

Now that we've deconstructed the problem and even come up with our own expression, let's look at the options provided and see which one matches our logic. We need to find the expression that represents the total cost 'cc' as a function of the number of boxes 'bb', where the first box is $4 and each additional box gets a $1 discount (making them 3each).Ourderivedexpressionwas′3 each). Our derived expression was 'c = 4 + (b - 1) * 3′.Let′ssimplifythis:′'. Let's simplify this: 'c = 4 + 3b - 3′,whichfurthersimplifiesto′', which further simplifies to 'c = 3b + 1

.

Let's analyze the given options:

Why c=3b+1c = 3b + 1 Works

Let's really hammer home why option C, 'c=3b+1c = 3b + 1', is the correct answer, guys. We established that the first box costs $4, and every box after that costs 3.Whenwe′rethinkingaboutthetotalcost′3. When we're thinking about the total cost 'c′for′' for 'b′boxes,wecanviewitasfollows:imagine∗all∗′' boxes, we can view it as follows: imagine *all* 'b

boxes were priced at the discounted rate of $3. The total cost for that would be '3b3b'. However, this isn't quite right because the first box wasn't $3; it was $4. So, we're $1 short for that first box if we just use '3b3b'. To correct this, we need to add that extra 1backintothetotalcosttoaccountforthehigherpriceofthefirstbox.Hence,′1 back into the total cost to account for the higher price of the first box. Hence, 'c = 3b + 1 .

Let's look at it from another angle, using our initial breakdown: 'c=extCostoffirstbox+extCostofadditionalboxesc = ext{Cost of first box} + ext{Cost of additional boxes}'.

The cost of the first box is $4.

The number of additional boxes is 'b−1b - 1' (only if b>0b > 0).

The cost of each additional box is $3.

So, the cost of additional boxes is '$ (b - 1) * 3

.

Total cost 'cc' = 4+(b−1)∗34 + (b - 1) * 3.

Now, let's simplify this expression algebraically:

c=4+(b∗3)−(1∗3)c = 4 + (b * 3) - (1 * 3) c=4+3b−3c = 4 + 3b - 3 c=3b+(4−3)c = 3b + (4 - 3) c=3b+1c = 3b + 1

This algebraic simplification directly leads us to option C. It elegantly captures the pricing structure: the base cost for subsequent boxes is $3 (making up the '3b3b' part), and the '+1

is the adjustment for the first box being $1 more expensive than the subsequent ones. This function correctly represents the total cost regardless of how many boxes Barbara buys (as long as bb is a positive integer representing the number of boxes).

Conclusion: Mastering Functions Through Pen Purchasing

So there you have it, team! We've successfully navigated the world of function notation by tackling Barbara's pen-buying dilemma. We started with a scenario involving a base price and a discount on additional items, a common situation we encounter in everyday life. By carefully breaking down the cost structure – identifying the fixed price of the first item and the discounted price of subsequent items – we were able to construct an accurate mathematical expression. We saw that the key was to represent the number of additional boxes as 'b−1b - 1' and apply the discount accordingly. This led us to the expression 'c=4+(b−1)∗3c = 4 + (b - 1) * 3'. Through algebraic simplification, we arrived at the concise and elegant function 'c=3b+1c = 3b + 1', which perfectly matches option C. We also debunked the other options, showing why they failed to account for the specific conditions of the discount. Remember, problems like this aren't just about finding the right answer; they're about understanding why it's the right answer. Functions are powerful tools that allow us to model real-world relationships, and by practicing these types of problems, you build a stronger intuition for how to translate descriptions into mathematical equations. Keep practicing, keep questioning, and you'll master these concepts in no time. Happy problem-solving, everyone!