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ExtremeBDS [4]
2 years ago
8

A rectangular box has a square base with an edge length of x cm and a height of h cm. The volume of the box is given by V = x2h

cm3. Find the rate at which the volume of the box is changing when the edge length of the base is 12 cm, the edge length of the base is decreasing at a rate of 2 cm/min, the height of the box is 6 cm, and the height is increasing at a rate of 1 cm/min.
Mathematics
2 answers:
creativ13 [48]2 years ago
4 0

It is given in the question that

A rectangular box has a square base with an edge length of x cm and a height of h cm. The volume of the box is given by

V = x^2h cm^3

And the edge length of the base is 12 cm, the edge length of the base is decreasing at a rate of 2 cm/min, the height of the box is 6 cm, and the height is increasing at a rate of 1 cm/min.

Here we differentiate V with respect to t, and we use product rule, that is

V = 2xh(dx/dt)+ x^2(dh/dt)

Substituting the given values , we will get

[tex]dV/dt = 2(12)(6)(-2)+ 12^2(1)[/tex]

dV/dt = -288+144 = -144cm^3/min

So at that moment, the volume is decreasing at the rate of

144 cm^3/min

Archy [21]2 years ago
4 0

The volume is decreasing at the rate of \boxed{{\mathbf{144c}}{{\mathbf{m}}^{\mathbf{3}}}{\mathbf{/min}}} .

Further explanation:

Given:

It is given that a rectangular box a square base with a length of the side is x{\text{ cm}}  and has a height of h{\text{ cm}} .

Step by step explanation:

Step 1:

The volume of the rectangular box can be calculated as,

V=lbh

Here, l  is the length,  b is breadth,  h is the height.

Therefore, volume of the box can be calculated as,

V={x^2}h{\text{ c}}{{\text{m}}^{\text{3}}}

Step 2:

Now differentiate volume with respect to l  both sides by the use of product rule.

  \frac{{dV}}{{dt}}=2xh\left({\frac{{dx}}{{dt}}}\right)+{x^2}\left({\frac{{dh}}{{dt}}}\right)

It is given that the length of the base is 12{\text{ cm}}  and it decreasing at the rate of .

The height of the box is 6{\text{ cm}}  and the height is increasing at the rate of 1{\text{ cm/min}} .

Therefore, the values are x=12,h=6,\frac{{dx}}{{dt}}=-2{\text{ and}}\frac{{dh}}{{dt}}=1

Now substitute the values  x=12,h=6,\frac{{dx}}{{dt}}=-2{\text{ and}}\frac{{dh}}{{dt}}=1 in the differential equation.

\begin{gathered}\frac{{dV}}{{dt}}=2xh\left({\frac{{dx}}{{dt}}}\right)+{x^2}\left({\frac{{dh}}{{dt}}}\right)\hfill\\\frac{{dV}}{{dt}}=2\left({12}\right)\left(6\right)\left({-2}\right)+{\left({12}\right)^2}\left(1\right)\hfill\\\frac{{dV}}{{dt}}=-144{\text{c}}{{\text{m}}^{\text{3}}}/\min\hfill\\\end{gathered}

Therefore, it can be seen that the volume is decreasing as the value of \frac{{dV}}{{dt}}  involves negative sign.

Thus, the volume is decreasing at the rate of 144c{m^3}/\min .

Learn more:  

  • Learn more about the function is graphed below <u>brainly.com/question/9590016 </u>
  • Learn more about which undefined geometric term is described as a location on a coordinate plane that is designated by an ordered pair, (x, y)? distance line plane point <u>brainly.com/question/2876945</u>
  • Learn more about midpoint of the segment <u>brainly.com/question/3269852 </u>

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Application of derivative

Keywords: side, length, square, base, differentiate, decreasing, increasing, rectangular box, respect, product rule, height, rate, volume of the box

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