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zheka24
15 days ago
9

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

Mathematics
1 answer:
Inessa [3.9K]15 days ago
5 0

Answer:

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

The variable x lies within the interval of all positive real numbers less than 5 cm.

Detailed solution:

Problem statement:

Determine the volume of the open-topped box as a function of the side length x (in centimeters) of the square cutouts.

Refer to the provided diagram for clarity.

Define:

x → length in centimeters of each square cutout side

The volume of the box with open top can be written as:

V=LWH

Given this, we have:

L=(10-2x)\ cm

W=(10-2x)\ cm

H=x)\ cm

By substitution:

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

Determine the domain of x:

Because:

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

Therefore:

Domain is the interval (0,5)

That means all real numbers strictly greater than zero and less than 5 cm are valid for x.

Hence, the volume V as a function of x is:

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

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Answer:

The main aspects are

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0.5ft³/min = filling rate

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Step-by-step breakdown:

Chiang is in the process of filling a 50ft container with water at a speed of 0.5 ft/min.

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