The perimeter of a rectangular piece of metal is 30 centimeters. The area is 54 square centimeters. What are the dimensions of the piece?

Answer 1

Hi there,

This sounds like more of a pure algebra problem than a SolidWorks one - you'd probably just have an easier time doing this by hand, as doing it in SolidWorks would mostly be trial and error. It sounds like you've been given these 3 pieces of information to solve for:

1.) The geometry shape is a rectangle

2.) The area is 54 cm²

3.) The perimeter is 30 cm

So based off of these points, you could use a system of equations to solve for the dimensions of the rectangle. Let's define our variables like this:

x = width of the rectangle

y = length of the rectangle

The area of the rectangle would just be the width multiplied by the length, so here is our first equation:

x*y=54

The perimeter is the sum of the lengths of all 4 sides of the rectangle. Since this is a rectangle, we know that 2 sides will have a length equal to the x variable, and 2 sides will have a length equal to the y variable (totaling 4 sides). That gives us our second equation:

2x+2y=30

So, now that we have 2 equations and 2 variables, we can solve for them. First, let's reduce that first equation down to a definition of x in terms of y. That might sound like gibberish, so let's just go thru the steps:

x*y=54

Now we divide both sides by y, which gives us this:

x=54/y

Alright, so now we have a definition of the x variable in terms of y. Now, we can sub this definition into the SECOND equation, and get a number value for y. Here we go:

2x+2y=30

2(54/y)+2y=30

108/y+2y=30

I'm going to multiply both sides by y in order to get rid of the y in the denominator:

108+2y²=30y

Then subtract 30y from both sides

2y²-30y+108=0

Now that we've reduced it to this format of ax²+bx+c=0, we can use the quadratic equation to solve for it. That gives us this:

y = (30 ± √(b²-4ac))/2a

y = (30 ± √(900-864))/4

y = (30 ± 6)/4

y = 6 or y = 9

So it'll have to be one of those 2 options. Let's give them a test run by subbing those values into the equations again

y=6

x=54/y

x=54/6

x=9

If we run with that definition on the first equation:

x*y=54

(9)*(6)=54

That looks good. On the second equation:

2x+2y=30

2(9)+2(6)=30

18+12=30

That also works out. Note that we could also reverse the values - by having y=9 and x=6, this would all still work out. So no matter how you slice it, the dimensions of the plate are 9 cm by 6 cm. Hope that helps!