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I was solving this problem from my algebra textbook:

A basketball court measures $25m \times 15m$. The court is surrounded by a row of benches that is the same width on all sides. If the row of benches has total area of $325m$$^2$, find the dimensions of the space needed for the court and benches. (Round answers to one decimal place if necessary)

I tried to solve the problem as follows:

Since we know the length of the benches we can call the width as $x$. Therefore we get the expression: $2(15x) + 2(25x) = 325$ and then you simply solve for $x$.

However, my answer was incorrect. Can somebody please help me solve the problem?

InanimateBeing
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Jaireet Chahal
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  • whats the given answer? –  Jul 24 '22 at 15:38
  • @HaHa when I asked my teacher about it they responded by saying "After you have your area formula you expand it to get a quadratic equation. Now do the quadratic formula to figure out what your root is, then sub in that root into the perimeter equation P = 2(25+2x)+2(15+2x) to get your perimeter and dimensions. " – Jaireet Chahal Jul 24 '22 at 15:40
  • Where did you get the equation you solved? The length of the space including the benches is $(25 + 2x)$m, – Ethan Bolker Jul 24 '22 at 15:41
  • @EthanBolker well the question tells us that the area of the benches is 325 metres squared. So my first instinct was to create an equation that modeled this. If we write the width of the benches as x then this means their length would be 25 and 15. Since area is length times width I knew that it would be 15x and 25x. Then I multiplied by 2 because there are 4 sides on a rectangle. – Jaireet Chahal Jul 24 '22 at 15:47
  • Your instinct to create an equation was right but you did not create the right equation. Draw a picture showing where the benches are added outside the court and label all the dimensions using numbers and $x$'s. – Ethan Bolker Jul 24 '22 at 15:50
  • @EthanBolker I originally drew a diagram and I attached it with the post. Please let me know if you cannot see it. – Jaireet Chahal Jul 24 '22 at 15:54
  • @JaireetChahal What means 1Sm_ ? – miracle173 Jul 24 '22 at 16:20
  • @miracle173 That is the label of 15 meters. I apologize for my poor handwriting – Jaireet Chahal Jul 24 '22 at 16:25
  • "The court is surrounded by a row of benches" : This arguably means that there are benches in the 4 corners, where the length of the basketball court meets its width. The dimension of each of these 4 corners of benches, which you have omitted in your calculations, is $x \times x = x^2.$ The question is: what is the problem composer's intent, re the word surrounded ? – user2661923 Jul 24 '22 at 17:22
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JaireetChahal I must say how you tackled the problem is ingenuous. But the classrooms are another matter :-\ .

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If above were the case, then the way you solved would have given the correct value of $x$. Sadly it is not. Though, given the wording of the question, it should be valid.

Let's look at how you are required to see the problem.

Pic2

Now, you should be able to work out the value of $x$ yourself and your attempt would be highly appreciable.

If you are still unable to get correct value of $x$ then look below:

Let the width of benches be $x$ metres.

We are given area of the red region as $325m$$^2=$ $(\text {Area of big rectangle})-(\text{Area of smaller rectangle})$.

$\Rightarrow 325 = (15+2x)(25+2x)-(15)(25)$

$\Rightarrow 325 = 4x^2+80x$

$\Rightarrow x=-10+\frac {5\sqrt { 29}}2=3.5 \text { (Approx.)}$

So, your classroom expects you to report $x$ as $3.5m$ instead of $4.1m$ as you were getting.

InanimateBeing
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