A shed has dimensions of 12m in length and 5 m in width. Both the length and width are increased by the same amount in order to increase the floor area by more than double the original area. What is the amount of the increase in length?
(Ok so this is the question, Im thinking that the answer is 24 but my teacher is giving a bunch of marks if I get it right, now I'm thinking it can't be that easy and there must be more to it right? Please help me)
original floor area = 12*5 = 60 m^2
let new length be 12+x
let new width be 5+x
(12+x)(5+x) = 120 , (double the original area)
60 + 17x + x^2 = 120
x^2 + 17x - 60 = 0
(x - 3)(x + 20) = 0
x = 3 or x = -20 , but we can't have a negative side, so
x = 3
check:
new length = 12+3 = 15
new width = 5 + 3 = 8
new area = 15*8 = 120 , which is double the original 60
YEAHH
To solve this problem, we need to determine the amount by which the length and the width need to be increased in order to more than double the original floor area of the shed.
Let's start by finding the original floor area of the shed. The formula to calculate the area is:
Area = length × width
Substituting the given dimensions, we have:
Original Area = 12m × 5m = 60m²
To determine the amount by which the length and width need to be increased, let's denote the increase as x.
Now, let's calculate the new dimensions after the increase:
New Length = 12m + x
New Width = 5m + x
The new area can be calculated using the same formula:
New Area = New Length × New Width
According to the problem, the new area should be more than double the original area. Therefore, we can write the following inequality:
New Area > 2 × Original Area
Substituting the expressions for the new area and the original area, we get:
(New Length × New Width) > 2 × (Original Length × Original Width)
(12m + x)(5m + x) > 2 × 60m²
Now, we can expand the inequality and solve for x:
(60m² + 12mx + 5mx + x²) > 120m²
60m² + 17mx + x² > 120m²
Rearranging the equation:
x² + 17mx + (60m² - 120m²) > 0
x² + 17mx - 60m² > 0
Now, we can factor the left side of the inequality:
(x + 20m)(x - 3m) > 0
To find the values of x that satisfy the inequality, we can set each factor separately equal to zero:
x + 20m = 0 or x - 3m = 0
Solving for x in each equation gives us:
x = -20m or x = 3m
Since we cannot have negative dimensions, the only valid solution is:
x = 3m
Therefore, the amount by which the length needs to be increased is 3 meters.
To find the amount of increase in length, we first need to calculate the original area of the shed.
The original area of the shed is given by the formula:
Area = length x width
Substituting the given values:
Original area = 12m x 5m = 60 square meters
To increase the floor area by more than double the original area, the new area should be more than 2 times the original area.
Let's assume that both the length and width are increased by "x" meters.
The new length would be 12m + x, and the new width would be 5m + x.
The new area would be:
New area = (12m + x) * (5m + x)
We know that the new area should be more than 2 times the original area, so we can set up the inequality:
New area > 2 * Original area
(12m + x) * (5m + x) > 2 * 60
Simplifying this inequality, we get:
(12 + x)(5 + x) > 120
Expanding and rearranging the terms, we have:
60 + 12x + 5x + x^2 > 120
x^2 + 17x + 60 > 120
x^2 + 17x - 60 > 0
Now we need to solve this quadratic inequality. We can factorize it as:
(x + 20)(x - 3) > 0
To solve this inequality, we consider the sign of each factored expression:
x + 20 > 0 AND x - 3 > 0
x > -20 AND x > 3
Since the value of "x" needs to be positive, the minimum value satisfying both conditions is x > 3.
Therefore, the amount of increase in length is greater than 3 meters. However, we cannot determine an exact value for "x" without further information, as there are multiple potential solutions that would satisfy the inequality.