a poster is 24 cm taller than it is wide. it if is mounted on a piece of cardboard so that there is a 6cm border on all sides and if the area of the border alone is 720 cm squared, what are the dimensions of the poster
To solve this problem, let's break it down into steps:
Step 1: Understand what we are looking for.
We want to find the dimensions (width and height) of the poster.
Step 2: Assign variables.
Let's assign a variable to the width of the poster. We'll call it "x." Since the poster is 24 cm taller than it is wide, the height of the poster will be x + 24 cm.
Step 3: Calculate the dimensions including the borders.
To find the dimensions of the poster, we need to consider the 6 cm border on each side. This means the actual width of the poster, including the borders, will be (x + 6 + 6) cm, and the actual height will be (x + 24 + 6 + 6) cm.
Step 4: Calculate the area of the border.
The area of the border can be calculated by subtracting the area of the poster itself from the total area. The total area of the cardboard with the borders is given as 720 cm^2.
Area of the border = Total area - Area of the poster
Step 5: Calculate the area of the poster.
The area of the poster can be calculated by multiplying its width by its height.
Area of the poster = Width * Height
Step 6: Set up the equation.
Now, let's set up the equation using the information we have:
720 = (x + 12) * (x + 36)
Step 7: Solve the equation.
We can solve this quadratic equation for x by expanding and simplifying:
720 = x^2 + 48x + 432
Now, rearrange the equation to bring everything to one side:
x^2 + 48x + 432 - 720 = 0
Simplify further:
x^2 + 48x - 288 = 0
Step 8: Solve the quadratic equation.
We can solve this quadratic equation by factoring or using the quadratic formula. Let's solve it by factoring:
(x + 72)(x - 4) = 0
Setting each factor to zero gives us two possible solutions:
x + 72 = 0 or x - 4 = 0
Solving for x in each case:
x = -72 or x = 4
Since the width cannot be negative, we discard x = -72 as an extraneous solution.
Step 9: Calculate the dimensions.
The width of the poster, x, is equal to 4 cm. So:
Width = x = 4 cm
Height = x + 24 = 4 + 24 = 28 cm
Therefore, the dimensions of the poster are 4 cm (width) and 28 cm (height).