The width of a rectangular painting is 3inches more than twice the height .It is frames in a 2.5inches wide frames. what is the expression for the combined are of the painting and frame? Factorise the expression if it is possible..
w = 2h+3
total w = w+5
total h = h+5
A = (w+5)(h+5)
= w h + 5w + 5h + 25
= (2h+3)h + 5(2h+3) + 5h +25
= 2h^2 +3h +10h +15 +5h + 25
= 2h^2 + 18h + 40
= 2(h^2 + 9 h + 20)
=2(h+4)(h+5)
Why did the painting go on a diet? Because it wanted to fit in its narrow frame! 😄
To find the expression for the combined area of the painting and frame, let's assume the height of the painting is "h" inches.
According to the given information, the width of the painting is 3 inches more than twice the height. So, the width would be (2h + 3) inches.
To calculate the area of the painting, we multiply the height by the width:
Area of the painting = h * (2h + 3)
Now, let's add the width of the frame on both sides of the painting. Since the frame is 2.5 inches wide, we need to add 2 * 2.5 = 5 inches to both the width and height.
Therefore, the combined area of the painting and frame would be:
Combined area = (h + 5) * (2h + 3 + 5)
= (h + 5) * (2h + 8)
Let's factorize the expression:
Combined area = 2h^2 + 8h + 10h + 40
= 2h^2 + 18h + 40
So, the expression for the combined area of the painting and frame is 2h^2 + 18h + 40.
Let's denote the height of the painting as h inches.
According to the given information, the width of the painting is 3 inches more than twice the height. Expressing this mathematically, we have:
Width = 2h + 3
To calculate the area of the painting, we multiply the width by the height:
Area of the painting = Width x Height
Area of the painting = (2h + 3) * h
Area of the painting = 2h^2 + 3h
Since the frame surrounds the painting on all sides, we need to add the width of the frame to both the width and height of the painting.
Width of frame = 2.5 inches
New width (including frame) = Width + (2 x Width of frame)
New width (including frame) = (2h + 3) + (2 x 2.5)
New width (including frame) = 2h + 3 + 5
New width (including frame) = 2h + 8
New height (including frame) = Height + (2 x Width of frame)
New height (including frame) = h + (2 x 2.5)
New height (including frame) = h + 5
The combined area of the painting and frame is given by:
Combined area = New width x New height
Combined area = (2h + 8) * (h + 5)
Combined area = 2h^2 + 18h + 40
The expression for the combined area of the painting and frame is 2h^2 + 18h + 40.
This expression cannot be factorized further.
To find the expression for the combined area of the painting and frame, we need to break down the problem into smaller steps:
Step 1: Assign variables
Let's assign variables to represent the height and width of the painting. We'll use "h" for the height and "w" for the width.
Step 2: Translate the given information into equations
The width of the painting is 3 inches more than twice the height. We can express this as:
w = 2h + 3
The width of the frame is given as 2.5 inches. Since there are two frames (top/bottom and left/right), we need to add the frame width twice to both the height and width:
Total height with frame = h + 2(2.5)
Total width with frame = w + 2(2.5)
Step 3: Find the combined area
The combined area of the painting and the frame is the product of the total height and the total width:
Combined area = (h + 2(2.5))(w + 2(2.5))
Step 4: Simplify the expression
Now, let's substitute the value of w from the first equation into the combined area expression:
Combined area = (h + 2(2.5))(2h + 3 + 2(2.5))
Simplifying further:
Combined area = (h + 5)(2h + 3 + 5)
Combining like terms:
Combined area = (h + 5)(2h + 8)
Step 5: Factorize the expression
The expression (h + 5)(2h + 8) cannot be further factorized because the terms inside the brackets do not share any common factors.
So, the expression for the combined area of the painting and frame is (h + 5)(2h + 8).