a rectangle is 4 times as long as it is wide. a second triangle is 5 centimeters longer and 2 centimeters wider than the first. the area of the second rectangle is 270 square centimeters greather than the first. what are the dimensions of the original rectangle? how would i solve this question?

Question

a rectangle is 4 times as long as it is wide. a second triangle is 5 centimeters longer and 2 centimeters wider than the first. the area of the second rectangle is 270 square centimeters greather than the first. what are the dimensions of the original rectangle? how would i solve this question?

Answer 1

first triangle : x by 4x

second triangle : 4x+5 by x+2

area of first = x(4x)
area of second = (4x+5)(x+2)

so (4x+5)(x+2) - x(4x) = 270

solve for x, you should get x=20

(check: original is 20 by 80, Area = 1600
second is 22 by 85, Area = 1870)
difference in area = 1870-1600 = 270

Answer 2

To solve this question, you need to set up a system of equations based on the given information and then solve it. Let's break down the problem step-by-step:

Step 1: Define the variables
Let's say the width of the original rectangle is "w" centimeters. Since the length of the rectangle is 4 times its width, the length would be 4w centimeters.

Step 2: Set up the equations
According to the problem, the length of the second triangle is 5 centimeters longer than the first. Since the first triangle's length is 4w, the second triangle's length would be (4w + 5).

Similarly, the width of the second triangle is 2 centimeters wider than the first, so it would be (w + 2).

Step 3: Calculate the areas
The area of a rectangle is given by the formula: Area = Length x Width. Thus, the area of the first rectangle is (4w x w), which is equal to 4w² square centimeters.

The area of the second rectangle is ((4w + 5) x (w + 2)), which is equal to (4w² + 13w + 10) square centimeters.

According to the problem, the area of the second rectangle is 270 square centimeters greater than the first rectangle. So, the equation becomes:
(4w² + 13w + 10) = (4w² + 270)

Step 4: Simplify and solve
To solve the equation, we need to make it an equation equal to zero. Subtracting (4w²) from both sides and simplifying, we get:
13w + 10 = 270

Subtracting 10 from both sides, we get:
13w = 260

Dividing both sides by 13, we find:
w = 20

Now, substituting this value of "w" back into the original equation to find the length, we have:
Length = 4w
Length = 4 * 20
Length = 80

Therefore, the dimensions of the original rectangle are:
Width = 20 centimeters
Length = 80 centimeters