Joanne has a garden that is 9 feet long by 6 feet wide. She wants to increase the dimensions of the garden by the same distance each way and have the area of the new garden be five times the area of the present garden. How many feet should she add to both the width and the length?
My answer: add 9ft to each side
Your answer is correct.
To solve this problem, let's start by finding the current area of the garden.
The current garden dimensions are:
Length = 9 feet
Width = 6 feet
The current area is calculated by multiplying the length and the width:
Area = Length * Width = 9 ft * 6 ft = 54 square feet
Now, let's determine the desired area of the new garden.
Joanne wants the area of the new garden to be five times the area of the current garden:
Desired Area = 5 * Current Area = 5 * 54 square feet = 270 square feet
To find the new dimensions of the garden, we need to add the same distance to both the length and the width.
Let's assume that Joanne adds x feet to both the length and the width. Therefore, the new dimensions will be:
Length + x
Width + x
Now, let's set up an equation using the desired area and the new dimensions:
Desired Area = (Length + x) * (Width + x)
Substituting the values we have:
270 square feet = (9 ft + x) * (6 ft + x)
To solve this equation, we can use the distributive property and then simplify:
270 square feet = (9 * 6) ft^2 + 9x + 6x + x^2
270 square feet = 54 ft^2 + 15x + x^2
Rearranging the terms and setting the equation to zero, we get a quadratic equation:
x^2 + 15x + 54 ft^2 - 270 ft^2 = 0
x^2 + 15x - 216 ft^2 = 0
Now we can factor or use the quadratic formula to solve for x. Factoring gives us:
(x - 9)(x + 24) = 0
Setting each factor to zero, we have two possibilities:
x - 9 = 0 or x + 24 = 0
x = 9 or x = -24
Since we are looking for a positive value for x (as it represents the added distance), we discard the negative solution. Therefore, the added distance is x = 9 feet.
Therefore, Joanne should add 9 feet to both the width and the length of her garden to have the area of the new garden be five times the area of the present garden.
To find out the increase she needs to make to both the width and the length, we can first calculate the area of the present garden. The area of a rectangle is found by multiplying its length by its width.
The present garden has a length of 9 feet and a width of 6 feet, so its area is 9 feet * 6 feet = 54 square feet.
Now, she wants the area of the new garden to be five times the area of the present garden, which means the new garden should have an area of 5 * 54 square feet = 270 square feet.
To find the dimensions of the new garden, we can set up an equation. Let x represent the increase in both the width and the length.
The length of the new garden will be 9 ft + x, and the width will be 6 ft + x. Multiplying these dimensions together should give us the desired area.
(9 ft + x) * (6 ft + x) = 270 square feet
Expanding this equation, we get:
54 + 15x + x^2 = 270
Rearranging and simplifying:
x^2 + 15x - 216 = 0
Now, we can solve this quadratic equation to get the value of x. Once we have the value of x, we will know how many feet she should add to both the width and the length.
You can use factoring, completing the square, or the quadratic formula to solve the equation.