Chapter 8.3, Problem 1MP

(a)

To determine

To find: The polynomial that represents the area of Kelly’s plot.

Answer to Problem 1MP

  $x^2+8x+12$is the polynomial that represents area of Kelly’s plot

Explanation of Solution

Given Information:

  

Formula Used:

Area of a rectangle of length l and width b is lb

Calculation:

Length of Kelly’s plot, l = $x+6$

Width of Kelly’s plot, b = $x+2$

Length and width of Kelly’s plot are binomials (since it contains two terms)

Therefore the area is the product of two binomials.

Area of Kelly’s plot, $A_1=lb$

  $=\left(x+6\right)\left(x+2\right)$

  $=x\left(x+2\right)+6\left(x+2\right)$

  $=x^2+2x+6x+12$

  $=x^2+8x+12$

(b)

To determine

To find: The polynomial that represents the area of Roberto’s plot.

Answer to Problem 1MP

  $x^2+9x+18$ is the polynomial that represents area of Roberto’s plot

Explanation of Solution

Given Information:

  

Formula Used:

Area of a rectangle of length l and width b is lb

Calculation:

Length of Roberto’s plot, l = $x+6$

Width of Roberto’s plot, b = $x+3$

Length and width of Roberto’s plot are binomials (since it contains two terms)

Therefore the area is the product of two binomials.

Area of Roberto’s plot, $A_2=lb$

  $=\left(x+6\right)\left(x+3\right)$

  $=x\left(x+3\right)+6\left(x+3\right)$

  $=x^2+3x+6x+18$

  $=x^2+9x+18$

(c)

To determine

To check: the products found in (a) and (b) are correct by applying a value for x.

Explanation of Solution

Given Information:

Kelly’s plot Length = $x+6$

Breadth= $x+2$

Roberto’s plot Length = $x+6$

Breadth = $x+3$

Proof:

Let’s assume that $x=1$

For Kelly’s plot Length = 7

Breadth= 3

Area = $7\times 3=21$

Now let’s compute the area using $x^2+8x+12$ for $x=1$

  $\begin{array}{c}{A}_{x=1}=1^2+8\cdot 1+12\\ =9+12\\ =21\end{array}$

Which is same as $l\cdot b$

Roberto’s plot Length = 7

Breadth= 4

Area = $7\times 4=28$

Now let’s compute the area using $x^2+9x+18$ for $x=1$

  $\begin{array}{c}{A}_{x=1}=1^2+9\cdot 1+18\\ =10+18\\ =28\end{array}$

Which is same as $l\cdot b$

∴The polynomials found in (a) and (b) are correct.

(d)

To determine

To check: whether the area of Roberto’s plot is greater than Kelly’s plot.

Answer to Problem 1MP

Area of Roberto’s plot is greater than that of Kelly’s plot.

Explanation of Solution

Given Information:

The area of Roberto’s plot is $x^2+9x+18$

The area of Kelly’s plot is $x^2+8x+12$

Proof:

From part (c)

For $x=1$

The area of Roberto’s plot = 28 sq.ft

The area of Kelly’s plot = 21 sq.ft

That is, for $x=1$

Area of Roberto’s plot is greater than that of Kelly’s plot……. (1)

Since the area is true for all values of x, (1) is true for all values of x