Calculate Area of House Shape Using Expressions 12x+9 and x+2y

Question

The height of the house in the drawing is 12x+9 12x+9

Whilst the width of the house x+2y x+2y

Given that the ceiling height is half the height of the square section.

Express the area of the house shape in the drawing :

Video Solution

Solution Steps

00:00 Express the area of the house shape using X and Y
00:06 Draw the two heights
00:13 The sum of the heights is the height of the shape
00:25 The triangle's height is half the square's height according to the given data
00:30 Substitute in the height value according to the given data
00:43 Multiply by the reciprocal fraction in order to isolate the square's height
00:57 Divide the brackets by 3
01:04 Open the brackets properly, multiply by each factor
01:09 This is the square's height
01:12 Substitute in this value in order to find the triangle's height
01:18 Open the brackets properly, multiply by each factor
01:22 This is the triangle's height
01:28 These are the heights
01:34 Apply the formula for calculating the area of a triangle
01:41 (height x base) divided by 2
01:44 W size according to the given data
01:49 Substitute this into the equation and solve to find the triangle's area
01:56 Open the brackets properly, multiply each factor by each factor
02:16 This is the triangle's area
02:19 Now let's calculate the square's area
02:26 Side x side
02:34 Substitute in the appropriate values and proceed to solve to find the square's area
02:44 Open the brackets properly, multiply each factor by each factor
03:01 This is the square's area
03:08 Continue calculating the triangle's area

Step-by-Step Solution

Let's draw a line in the middle of the drawing that divides the house into 2

Meaning it divides the triangle and the rectangular part.

The 2 lines represent the heights in both shapes.

If we connect the height of the roof with the height of the rectangular part, we obtain the total height.

Let's insert the known data in the formula:

12hsquare+hsquare=12x+9 \frac{1}{2}h_{\text{square}}+h_{square}=12x+9

32hsquare=12x+9 \frac{3}{2}h_{\text{square}}=12x+9

We'll multiply by two thirds as follows:

hsquare=2(12x+9)3=2(4x+3) h_{\text{square}}=\frac{2(12x+9)}{3}=2(4x+3)

hsquare=8x+6 h_{\text{square}}=8x+6

If the height of the triangle equals half the height of the rectangular part, we can calculate it using the following formula:

htriangle=12(8x+6)=4x+3 h_{\text{triangle}}=\frac{1}{2}(8x+6)=4x+3

Now we can calculate the area of the triangular part:

(x+2y)×(4x+3)2=4x2+3x+8xy+6y2=2x2+1.5x+4xy+3y \frac{(x+2y)\times(4x+3)}{2}=\frac{4x^2+3x+8xy+6y}{2}=2x^2+1.5x+4xy+3y

Now we can calculate the rectangular part:

(x+2y)×(8x+6)=8x2+6x+16xy+12y (x+2y)\times(8x+6)=8x^2+6x+16xy+12y

Now let's combine the triangular area with the rectangular area to express the total area of the shape:

S=2x2+1.5x+4xy+3y+8x2+6x+16xy+12y S=2x^2+1.5x+4xy+3y+8x^2+6x+16xy+12y

S=10x2+20xy+7.5x+15y S=10x^2+20xy+7.5x+15y

Answer

3x2+8xy+112x+4y2+3y 3x^2+8xy+1\frac{1}{2}x+4y^2+3y