# The perimeter of a rectangle is 148mm. The height is fourteen more than three times the base. What is the area?

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### 2 Answers

The perimeter is two times the base plus two times the height and equals 148.

2H+2B=148

The height is 14 more than three times the base.

H=14+3B

Write as a system of two equations with two unknowns.

H+B=74 (1)

H-3B=14 (2)

Multiply (1) by -1 and add to (2) to solve by elimination.

`-H-B=-74`

`H-3B=14`

----------------------

`-4B=-60`

`B=15`

Substitute 15 for B in (1) and solve for H.

`H=74-15=59`

To verify substitute 59 for H and 15 for B in (2).

59-(3)(15)=14

59-45=14

Calculate the area of the rectangle.

`Area=BxH`

Substitute 15 for B and 59 for H.

`Area=15x59=885`

**Thus the area of the rectangle is 885 mm.**

Before finding the area, the first task is finding the height and the base of the rectangle. The question states that the height is "fourteen more than three times the base". Let's set the variable b to represent base and h to represent height. Now, translating the previous statement from English to math, we get h = 14 + 3b. This puts one variable in terms of the other (h in terms of b).

Now let's think about the equation for the perimeter of a rectangle. The perimeter of any polygon can be found by adding the lengths of all the sides together. However, since a rectangle has two sets of matching sides, the perimeter P equals 2b + 2h. Let's plug everything we know so far into this equation:

P = 2b + 2h now substitute in the previous expression for h

P = 2b + 2(14 + 3b) now multiply it out and put in the value for P

148 = 2b + 28 + 6b now combine like terms and subtract 28 from both sides

120 = 8b now divide by 8 to find the value for b

15 = b plug in this value into the expression to find h

h = 14 + 3b = 14 + 3(15) = 14 + 45 = 59

We know the values for b and h (15 and 59, respectively). Adding in the units here, we get that the base is 15 mm and the height is 59 mm. To find the area, we will just multiply the base and the height, which gives us:

A = 15mm * 59mm = **885 mm^2** (keep in mind this is millimeter squared since you are multiplying two millimeter values!)