Exercise 27 Page 205

a The smaller rectangles that make up the bigger rectangle are a product of that rectangle's length and width. Since the left-most rectangle has an area of

x y

and its horizontal side is

x,

we know that its vertical side must be

y .

x y = x \cdot y

This also means that the rectangle with an area of

y^{2}

has a vertical side of

y .

With this information, we can find the area of the missing part.

By adding the expressions for the smaller rectangle's area, we can write an expression showing the area of the larger rectangle. To write the area as a product, we add the expressions along the bigger rectangle's horizontal side and multiply by its vertical side.

Sum : Product : A = x y + 3 y + y^{2} A = y (x + 3 + y)

Since both the sum and the product describe the same thing, the area of the big rectangle, we can equate them.

x y + 3 y + y^{2} = y (x + 3 + y)

b Like in Part A, we know that the smaller rectangle's area is a product of its sides. With this information we can identify the sides of the smaller rectangles.

We can now write expressions for the area of the two unknown rectangles.

3 (x) 3 (8) = 3 x = 24

Let's add this information to the diagram

Similar to Part A, we will create expressions showing the area of the big rectangle as a sum and as a product.

Sum : Product : A = x^{2} + 11 x + 24 A = (x + 8) (x + 3)

Since both the sum and the product describe the same thing — the area of the big rectangle — we can equate them.

x^{2} + 11 x + 24 = (x + 8) (x + 3)

Subchapter links

4.1.1 Line of Best Fit p.196-197

4.1.2 Residuals p.201-202

4.1.3 Upper and Lower Bounds p.205-206

4.1.4 Least Squares Regression Line p.210-213