The area of a rectangle of width ycm is 140cm^2. If the width is reduced by 2cm, the length increases by 3cm, and the area decreases to 136cm^2 to form an equation that enables you to determine the value of Y and hence, find the diagonal of the original rectangle
workings.
Area area= length length*width width
140-136=x 3*y-2
4=x 3*y-2
collect like terms
4-3 2=x*y
3=x*y
divide both sides by x
y=3/x.. Please am not sure am correct
y = the width
L = the length
A = the area of original rectangle
A = y * L = 140
y * L = 140 Divide both sides by y
L = 140 / y
New area :
( y - 2 ) ( L + 3 ) = 136
( y - 2 ) ( 140 / y + 3 ) = 136
y * 140 / y - 2 * 140 / y + 3 * y - 3 * 2 = 136
140 - 280 / y + 3 y - 6 = 136
- 280 / y + 3 y + 134 = 136 Subtract 134 to both sides
- 280 / y + 3 y + 134 - 134 = 136 - 134
- 280 / y + 3 y = 2 Multiply both sides by y
- 280 * y / y + 3 y * y = 2 * y
- 280 + 3 y ^ 2 = 2 y
3 y ^ 2 - 280 = 2 y Subtract 2 y to both sides
3 y ^ 2 - 280 - 2 y = 2 y - 2 y
3 y ^ 2 - 2 y - 280 = 0
The solutions are :
y = - 28 / 3 and y = 10
The width can't be negative so y = 10 cm
L = 140 / y = 140 / 10 = 14
L = 14 cm
The diagonal of the original rectangle :
d = sqroot ( y ^ 2 + L ^ 2 ) =
sqroot ( 10 ^ 2 + 14 ^ 2 ) =
sqroot ( 100 + 196 ) = sqrt ( 296 ) = 17.20465 cm
To find the value of y and hence the diagonal of the original rectangle, let's break down the problem into steps:
1. Given that the area of the rectangle is 140cm^2, we can write the equation: Area = length * width.
Substituting the width as ycm, the equation becomes 140 = length * y.
2. We are also given that when the width is reduced by 2cm and the length increases by 3cm, the area becomes 136cm^2.
Using the same equation, we get 136 = (length + 3) * (y - 2).
Now let's solve the equations to find the value of y:
1. Rearrange the first equation to get the length in terms of y:
length = 140 / y.
2. Substitute the expression for length in the second equation:
136 = (140 / y + 3) * (y - 2).
3. Expand the equation:
136 = (140 + 3y) / y * (y - 2).
4. Simplify by multiplying both sides by y:
136y = (140 + 3y) * (y - 2).
5. Expand and simplify the right side of the equation:
136y = 140y - 280 + 3y^2 - 6y.
6. Rearrange the equation to form a quadratic equation:
0 = 3y^2 - 6y - 140y + 280 + 136y.
7. Combine like terms:
0 = 3y^2 - 250y + 280.
8. Factor the quadratic equation:
0 = (3y - 20)(y - 14).
9. Set each factor equal to zero and solve for y:
3y - 20 = 0 or y - 14 = 0.
y = 20/3 or y = 14.
Since we are dealing with a length, the value of y cannot be negative. Therefore, y = 14.
Finally, to find the diagonal of the original rectangle, we can use the Pythagorean theorem. The length (l) and width (w) are given by l = 140/y and w = y. The diagonal (d) can be calculated as sqrt(l^2 + w^2).
Substituting the values, we get d = sqrt((140/14)^2 + 14^2) = sqrt(10^2 + 14^2) = sqrt(100 + 196) = sqrt(296), approximately equal to 17.20 cm.
Therefore, the diagonal of the original rectangle is approximately 17.20 cm.