Question

Please answer with clear explanation and the answer, I'm confused even tho I have done a similar question thank you

Answer 1

`\textsf{(i)} \quad (1)/(3)\left(3x-2y)(2x+3y)`

`\textsf{(ii)} \quad p = 3,\;\;q = 2`

`\textsf{(iii)} \quad x=5y`

Explanation:

Given expression:

`2x^2+(5)/(3)xy-2y^2`

Part (i)

To factor the given expression, first factor out the common term 1/3:

`\implies (1)/(3)\left(6x^2+5xy-6y^2\right)`

Rewrite the term 5xy as 9xy - 4xy:

`\implies (1)/(3)\left(6x^2+9xy-4xy-6y^2\right)`

Factor the first two terms and the last two terms inside the parentheses separately:

`\implies (1)/(3)\left(3x(2x+3y)-2y(2x+3y)\right)`

Factor out the common term (2x + 3y):

`\implies (1)/(3)\left(3x-2y)(2x+3y)`

Part (ii)

The volume of a cuboid is the product of its height and the length of each side of the base.

Given the cuboid has a height of 1/3m and that the length of each side of the base can be expressed as (px - qy) m or (qx + py) m, then the equation for the volume of the cuboid is:

`V=(1)/(3)(px-qy)(qx+py)`

Compare this with the factored equation from part (i). Therefore, the value of p and q is:

• p = 3
• q = 2

Part (iii)

As the base of the cuboid is square, the side lengths of the base are equal. Therefore, to express x in terms of y, equate the two expressions for the lengths of the side bases and isolate x.

`\implies 3x-2y=2x+3y`

`\implies 3x-2y-2x=2x+3y-2x`

`\implies x-2y=3y`

`\implies x-2y+2y=3y+2y`

`\implies x=5y`