Question

Mai wants to make an open top box by cutting out corners of a square piece of cardboard and folding up the sides. The cardboard is 10 cm by 10 cm. The volume V(x) in cubic cm of the open top box is a function of the side length x in cm of the square cutouts

Answer 1

Answer:

$V(x)=(4x^{3}-40x^{2}+100x)\ cm^3$

The domain for x is all real numbers greater than zero and less than 5 com

Step-by-step explanation:

The question is

What is the volume of the open top box as a function of the side length x in cm of the square cutouts?

see the attached figure to better understand the problem

Let

x -----> the side length in cm of the square cutouts

we know that

The volume of the open top box is

$V=LWH$

we have

$L=(10-2x)\ cm$

$W=(10-2x)\ cm$

$H=x)\ cm$

substitute

$V(x)=(10-2x)(10-2x)x\\\\V(x)=(100-40x+4x^{2})x\\\\V(x)=(4x^{3}-40x^{2}+100x)\ cm^3$

Find the domain for x

we know that

$(10-2x) > 0\\10> 2x\\ 5 > x\\x < 5\ cm$

so

The domain is the interval (0,5)

The domain is all real numbers greater than zero and less than 5 cm

therefore

The volume of the open top box as a function of the side length x in cm of the square cutouts is

$V(x)=(4x^{3}-40x^{2}+100x)\ cm^3$

Answer 2

The volume of a shape is the amount of space in it;

The volume as a function of side length (x) is: $V(x)= 4x^3 - 40x^2 + 100x$

From the question, we have the dimension of the cardboard to be;

$Length = 10$

$Width = 10$

When side length (x) is removed from the cardboard, the dimension of the box becomes:

$Length = 10 - 2x$

$Width = 10 - 2x$

$Height = x$

So, the volume of the box is:

$Volume = Length \times Width \times Height$

Substitute known values

$Volume = (10 - 2x) \times (10 - 2x) \times x$

Expand

$Volume = (10 - 2x) \times (10x - 2x^2)$

Expand

$Volume = 100x - 20x^2 - 20x^2 + 4x^3$

$Volume = 100x - 40x^2 + 4x^3$

Rewrite as:

$Volume = 4x^3 - 40x^2 + 100x$

Express as a function

$V(x)= 4x^3 - 40x^2 + 100x$

Hence, the required function is: $V(x)= 4x^3 - 40x^2 + 100x$

Read more about volume functions at:

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