Razman wants to sell a rectangular cake at a fair. The length and width of the cake are $(x+6)$ cm and $(x+3)$ cm, respectively. (a) Express the area of the cake, $L$ cm$^2$, in terms of $x$. (b) Given that the area of the cake is 270 cm$^2$, find the value of $x$. (c) Safi bought the cake for her birthday party and wants to divide the cake among her 30 friends. Will all her friends get a piece of the cake if Safi gives 9 cm$^2$ to each person? Explain your answer.

1. Problem Description

Razman wants to sell a rectangular cake at a fair. The length and width of the cake are $(x+6)$ cm and $(x+3)$ cm, respectively.

(a) Express the area of the cake, $L$ cm$^2$, in terms of $x$.

(b) Given that the area of the cake is 270 cm$^2$, find the value of $x$.

(c) Safi bought the cake for her birthday party and wants to divide the cake among her 30 friends. Will all her friends get a piece of the cake if Safi gives 9 cm$^2$ to each person? Explain your answer.

2. Solution Steps

(a) The area of a rectangle is given by the formula:

$Area = Length \times Width$

In this case, $Length = (x+6)$ and $Width = (x+3)$.

Therefore, the area $L$ of the cake is:

$L = (x+6)(x+3)$

$L = x(x+3) + 6(x+3)$

$L = x^2 + 3x + 6x + 18$

$L = x^2 + 9x + 18$

(b) We are given that the area of the cake is 270 cm$^2$. Therefore,

$x^2 + 9x + 18 = 270$

$x^2 + 9x + 18 - 270 = 0$

$x^2 + 9x - 252 = 0$

We need to find two numbers that multiply to -252 and add up to

9. Those numbers are 21 and -

1

2. $x^2 + 21x - 12x - 252 = 0$

$x(x + 21) - 12(x + 21) = 0$

$(x - 12)(x + 21) = 0$

$x - 12 = 0$ or $x + 21 = 0$

$x = 12$ or $x = -21$

Since the length and width cannot be negative, $x$ must be positive. Therefore, $x = 12$.

(c) Safi wants to divide the cake among 30 friends, giving each person 9 cm$^2$.

The total area Safi wants to give away is:

$30 \times 9 = 270$ cm$^2$

The actual area of the cake is 270 cm$^2$ (given).

Since the total area Safi wants to give away is equal to the area of the cake, all her friends will get a piece of the cake.

3. Final Answer

(a) $L = x^2 + 9x + 18$

(b) $x = 12$

(c) Yes, all her friends will get a piece of the cake because the total area Safi wants to distribute ($30 \times 9 = 270$ cm$^2$) is equal to the area of the cake (270 cm$^2$).