Correct to 1 significant figure, the area of a rectangle is 80 cm2. Correct to 2 significant figures, the length of the rectangle is 12 cm. Calculate the upper bound for the perimeter of the rectangle

Question

Correct to 1 significant figure, the area of a rectangle is 80 cm2. Correct to 2 significant figures, the length of the rectangle is 12 cm. Calculate the upper bound for the perimeter of the rectangle. Show your working clearly.

Answer 1

The problem is scarier than it actually is; a couple of math words have been stuffed in here to make it LOOK harder than it is.

UPPER BOUND: the largest number that the answer can be. 

SIGNIFICANT DIGITS: how many decimal places you are sure of in the number. When you round a number, all the places are exact EXCEPT the digit you Rounded to. 

What do we want: the PERIMETER (let’s call it P)

What do we know:

the AREA is 80.00

one side is 12.000 (let’s call it L)

let’s call the other side W

formula for PERIMETER is P = 2 (L + W)

formula for AREA is A = L • W

P = 2(L + W)

P = 2(12.000 + W)

we don’t know W, but can we find it?

A = L • W

80.00 = 12.000 • W

80.00 ÷ 12.000 = W (divide both sides by 12.000 to get “move” it away from W)

6.666666… = W [see the note below]

round it to 2 significant places:  

W = 6.667

P = 2(12.000 + W)

P = 2(12.000 + 6.667)

P = 2(18.667)

P = 37.334

the most significant digits our answer has is the smallest number of digits given…

the PERIMETER is 37.33.

There is only one answer. So it’s the Largest answer there is.

 PERIMETER = 37.33 cm (don’t forget the digits!)

NOTE: if you work in fractions instead of decimals, you are working with exact numbers. (2 divided by 3 is 2/3 EXACTLY, it’s only CLOSE to 0.667.). 

Answer 2

w = A/L

One boundary would be the largest possible value of "w", meaning the largest "A" and the smallest "L".

So w = 84.999999/11.5 = 7.3913 meaning P = 2(11.5) + 2(7.3913) = 37.7826

Another boundary would be the smallest "A" and largest "L" so w = 75/12.49999 = 6 so P = 12.49999(2) + 6(2) = 36.9999

As a reference, the nominal value of the perimeter is calculated when w = 80/12 or P = 12(2) + (80/12)(2) = 37.3333

So the upper bound is 37.7826 cm