the length of a rectangle is 8 cm more than its width. If the length is decreased by 9 and the width is tripled, the area is increased by 50%. What was the area of the original rectangle?
x=B,L=x+8, 3/2(x+8)(x)=(x+8-9)(3x),3/2(x(x)+8x)=3x(x-1), 3x(x)+24x=2(3x(x)+3x,factorisex,6x-6=3x+24,x=10,A=18(10),A=180
Are we supposed to decipher what all that is ?
Math is a precise language, and as such needs proper statements and presentation.
Sample:
let the width be x cm
then the length is x+8 cm
original area = x(x+8) cm^2
new width = 3x cm
new length = x+8 - 9 = x-1 cm
new area = 3x(x-1)
new area = 1.5(old area)
3x(x-1) = 1.5(x(x+8))
3x^2 - 3x = 1.5x^2 + 12x
1.5x^2 - 15x = 0
3x^2 - 30x = 0
3x(x - 10) = 0
3x = 0
x = 0 , not very likely
or
x = 10
original rectangle was 10 by 18 for an area of 180
new rectangle was 30 by 9 for an area of 270
Check:
What is 150% of 180? Why , sure enough it is 180
Let's solve this step-by-step:
Let's assume the width of the rectangle is "x" cm.
According to the given information, the length of the rectangle is 8 cm more than its width. So, the length would be "x + 8" cm.
The area of the rectangle is given by the formula: Area = Length × Width.
So, the original area of the rectangle would be:
Original Area = (x + 8) cm × x cm
Now, according to the second part of the question, when the length is decreased by 9 and the width is tripled, the area is increased by 50%. This means that the new area is 50% more than the original area.
Let's calculate the new length and width:
New Length = (x + 8 - 9) cm = (x - 1) cm
New Width = 3x cm
And the new area is:
New Area = (x - 1) cm × 3x cm = 3x^2 - 3x cm^2
Now, we can write the equation for the new area being 50% more than the original area:
New Area = Original Area + 50% of Original Area
(3x^2 - 3x) cm^2 = (x + 8) cm × x cm + 0.5 × [(x + 8) cm × x cm]
Simplifying the equation:
3x^2 - 3x = x^2 + 8x + 0.5x^2 + 4x
3x^2 - 3x = 1.5x^2 + 12x
Rearranging the terms:
1.5x^2 - 15x = 0
Factoring out x:
x(1.5x - 15) = 0
Solving for x, we get:
x = 0 (rejecting as width cannot be negative)
1.5x - 15 = 0
1.5x = 15
x = 10
So, the width of the original rectangle is 10 cm, and the length is x + 8 = 10 + 8 = 18 cm.
The original area of the rectangle is:
Original Area = (10 cm + 8 cm) × 10 cm = 18 cm × 10 cm = 180 cm^2.
Therefore, the area of the original rectangle is 180 cm^2.
To find the area of the original rectangle, we need to break down the problem into smaller steps.
Let's start by assigning variables to the width and length of the original rectangle.
Let's say the width of the original rectangle is "w" cm.
Then, according to the problem, the length will be 8 cm more than the width, which can be expressed as w + 8 cm.
The area of a rectangle is calculated by multiplying the length and the width, so the area of the original rectangle can be represented as:
Area = length × width
Now, let's move to the next step.
According to the problem, if the length is decreased by 9 cm, it becomes (w + 8) - 9 cm.
If the width is tripled, it becomes 3w cm.
With these new dimensions, the area of the modified rectangle is increased by 50%. This means the area of the modified rectangle is equal to 1.5 times the area of the original rectangle.
So, the equation for the area of the modified rectangle can be written as:
1.5 × Area = (w + 8 - 9) × (3w)
Now, let's solve this equation to find the value of w and ultimately the area of the original rectangle.
1.5 × Area = (w - 1) × (3w)
1.5 × Area = 3w² - 3w
3w² - 3w - 1.5 × Area = 0
We can now solve this quadratic equation to find the value of w using the quadratic formula.
w = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 3, b = -3, and c = -1.5 × Area.
Once we find the value of w, we can substitute it back into the equation for the area of the original rectangle:
Area = w × (w + 8)