Mark B. answered • 03/02/18
Tutor
New to Wyzant
PhD Candidate in Psychology: Experienced Math, Statistics, Tutor
Hello Harli,
So, what you are given in this problem are the original dimensions of a rectangle, but since neither of us know what those values are, we cannot really calculate the area for the rectangle, or can we? We are also told that when adding 2 units (whatever those units may be) to both the length and width of that rectangle, we will find that area to be 85 square units. So, let's start working the problem this way, okay? We know the formula for the area of a rectangle is, what?
A = L * W where A equals the area, L equals the length, and W equals the width. We good so far? Therefore, given the information from the problem the following equation emerges:
A = 5x * x <-------We are told the length is 5 times the width, so we use x for width and 5x for length
Now we are also told that increasing both length and width by 2 units will give us an area of 85 square units, right? So,
New area:
A = (5x +2)(x + 2) <------We merely added 2 units to our original expressions. Do you see it? Okay.
Remember the FOIL method for obtaining the product of two binomials: F equals the product of the first terms of each binomial, O equals the product of the outside terms of each binomial, I equals the product of the inside terms of each binomial, and L equals the product of the outer or outside terms of each binomial, right? Therefore:
5x2 + 10x + 2x + 4 = 85 <------We merely obtained the product of both terms when 2 units were added.
5x2 + 12x = 81 <------We subtracted 4 from both sides of the equation and added like terms.
5x2 + 12x - 81 = 0 <------Subtracted 81 from both sides of the equation and set to 0.
(5x + 27)(x - 3) = 0 <-------Factor the left side of the equation.
5x + 27 = 0 OR x -3 = 0 <------Set both factors to zero due to Principal of Zero Products. And now, solve.
x = -27/5 OR x = 3 <------Let's use 3 as the answer, because it is not as messy, okay?
Check our work:
(5(3) + 2)((3) + 2) = 85 <------We've only plugged in our value of x in the new length and width. Solve.
17 * 5 = 85 <------This, in fact, proves that our obtained value of 3 is correct when plugging the values into the equation for solving the area, right. But your problem asks for the ORIGINAL area.
Remember the original values we assigned of x and 5x due to that information provided? Using the same formula for the area we now can determine what the original area of the rectangle is, okay?
Since x is three:
A = L * W
5x * x
So, what you are given in this problem are the original dimensions of a rectangle, but since neither of us know what those values are, we cannot really calculate the area for the rectangle, or can we? We are also told that when adding 2 units (whatever those units may be) to both the length and width of that rectangle, we will find that area to be 85 square units. So, let's start working the problem this way, okay? We know the formula for the area of a rectangle is, what?
A = L * W where A equals the area, L equals the length, and W equals the width. We good so far? Therefore, given the information from the problem the following equation emerges:
A = 5x * x <-------We are told the length is 5 times the width, so we use x for width and 5x for length
Now we are also told that increasing both length and width by 2 units will give us an area of 85 square units, right? So,
New area:
A = (5x +2)(x + 2) <------We merely added 2 units to our original expressions. Do you see it? Okay.
Remember the FOIL method for obtaining the product of two binomials: F equals the product of the first terms of each binomial, O equals the product of the outside terms of each binomial, I equals the product of the inside terms of each binomial, and L equals the product of the outer or outside terms of each binomial, right? Therefore:
5x2 + 10x + 2x + 4 = 85 <------We merely obtained the product of both terms when 2 units were added.
5x2 + 12x = 81 <------We subtracted 4 from both sides of the equation and added like terms.
5x2 + 12x - 81 = 0 <------Subtracted 81 from both sides of the equation and set to 0.
(5x + 27)(x - 3) = 0 <-------Factor the left side of the equation.
5x + 27 = 0 OR x -3 = 0 <------Set both factors to zero due to Principal of Zero Products. And now, solve.
x = -27/5 OR x = 3 <------Let's use 3 as the answer, because it is not as messy, okay?
Check our work:
(5(3) + 2)((3) + 2) = 85 <------We've only plugged in our value of x in the new length and width. Solve.
17 * 5 = 85 <------This, in fact, proves that our obtained value of 3 is correct when plugging the values into the equation for solving the area, right. But your problem asks for the ORIGINAL area.
Remember the original values we assigned of x and 5x due to that information provided? Using the same formula for the area we now can determine what the original area of the rectangle is, okay?
Since x is three:
A = L * W
5x * x
5(3) * 3
15 x 3
15 x 3
45 square units <--------- the new area
I hope I have helped you and that I have explained everything in detail for you. Remember: When solving equations like these, you can draw from your toolbox which has provided you with some invaluable tools to solve your problem. I wish you a great Friday and weekend, and please feel free to ask any questions or provide any commentary below.
I hope I have helped you and that I have explained everything in detail for you. Remember: When solving equations like these, you can draw from your toolbox which has provided you with some invaluable tools to solve your problem. I wish you a great Friday and weekend, and please feel free to ask any questions or provide any commentary below.
