Question

Suppose you own a family bakeshop and you are to cut different sizes of rectangular box board to be used as a box for your business.The sizes of the box board you have are listed below: Box 1:The length is twice its width and the area is 338 sq.in.
Box 2:The length is 12 inches less than thrice its width and the area is 96 sq.in. Box 3: The perimeter of the box board is 80 in. and the area is 384 sq.in. process questions:
a.what quadratic equation represents the area of each piece of the box board?Write the equation in terms of the width of the box board
b. Write each quadratic equation formulated in item 1 in standard form. Then determine the values of a,b and c
c.Solve each quadratic equation using the quadratic formula.
d.Which of the solutions or roots obtained represents the width of each box board?Explain your answer.
e.What is the length of each piece of the box board?Explain how you arrived at your answer.

Answer 1

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Answer 2

Given:

Suppose you own a family bakeshop and you are to cut different sizes of rectangular box board to be used as a box for your business. The sizes of the box board you have are listed below:

Box 1:The length is twice its width and the area is 338 sq. in.

Box 2:The length is 12 inches less than thrice its width and the area is 96 sq. in.

Box 3: The perimeter of the boxboard is 80 in. and the area is 384 sq. in.

To Find:

a) what quadratic equation represents the area of each piece of the box board? Write the equation in terms of the width of the boxboard

b) Write each quadratic equation formulated in item 1 in standard form. Then determine the values of a, b and c

c) Solve each quadratic equation using the quadratic formula.

d) Which of the solutions or roots obtained represents the width of each box board? Explain your answer.

e) What is the length of each piece of the box board? Explain how you arrived at your answer.

Solution:

Let us start by calculating the dimensions of the 3 boxes one by one and use the quadratic formula whenever applicable

(a) Box 1

Let the breadth be x so the length will be 2x. Therefore,

$area=length*breadth\\338=2x*x\\x^2=169\\x^2-169=0$

Now compare it with the standard form of a quadratic equation and find the values of a,b,c and put in the Quadratic formula to find the roots

$ax^2+bx+c=0\\so,\\a=1\\b=0\\c=-169$

$x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} \\x=\frac{0\pm\sqrt{0+4*169} }{2}\\x=\pm13$

so taking 13

Hence, the length will be 26inch and the breadth will be 13inch.

(b) Box 2

Let the breadth be x so the length will be (3x-12). therefore,

$Area=l*b\\96=x*(3x-12)\\32=x^2-4x\\x^2-4x-32=0$

now using the quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} \\=\frac{4\pm\sqrt{16+128} }{2}\\=8,-4$

so taking 8

Hence the breadth will be 8inch and the length will be 12inch.

(c) Box 3

Let length be x and breadth be y

$1) x+y=40\\2) xy=384\\now,\\x+\frac{384}{x} =40\\x^2-40x+384=0$

Now using the Quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} \\=\frac{40\pm\sqrt{1600-1536} }{2} \\=24,16$

Hence, length and breadth both can be either 24inch or 16inch.