1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Crazy boy [7]
2 years ago
6

The radius of a sphere is increasing at a rate of 4 mm/s. how fast is the volume increasing when the diameter is 40 mm?

Physics
1 answer:
marin [14]2 years ago
7 0

Using <span>r </span> to represent the radius and <span>t </span> for time, you can write the first rate as:

<span><span><span><span>dr</span><span>dt</span></span>=4<span>mms</span></span> </span>

or

<span><span>r=r<span>(t)</span>=4t</span> </span>

The formula for a solid sphere's volume is:

<span><span>V=V<span>(r)</span>=<span>43</span>π<span>r3</span></span> </span>

When you take the derivative of both sides with respect to time...

<span><span><span><span>dV</span><span>dt</span></span>=<span>43</span>π<span>(3<span>r2</span>)</span><span>(<span><span>dr</span><span>dt</span></span>)</span></span> </span>

...remember the Chain Rule for implicit differentiation. The general format for this is:

<span><span><span><span><span>dV<span>(r)</span></span><span>dt</span></span>=<span><span>dV<span>(r)</span></span><span>dr<span>(t)</span></span></span>⋅<span><span>dr<span>(t)</span></span><span>dt</span></span></span> </span>with <span><span>V=V<span>(r)</span></span> </span> and <span><span>r=r<span>(t)</span></span> </span>.</span>

So, when you take the derivative of the volume, it is with respect to its variable <span>r </span> <span><span>(<span><span>dV<span>(r)</span></span><span>dr<span>(t)</span></span></span>)</span> </span>, but we want to do it with respect to <span>t </span> <span><span>(<span><span>dV<span>(r)</span></span><span>dt</span></span>)</span> </span>. Since <span><span>r=r<span>(t)</span></span> </span> and <span><span>r<span>(t)</span></span> </span> is implicitly a function of <span>t </span>, to make the equality work, you have to multiply by the derivative of the function <span><span>r<span>(t)</span></span> </span> with respect to <span>t </span> <span><span>(<span><span>dr<span>(t)</span></span><span>dt</span></span>)</span> </span>as well. That way, you're taking a derivative along a chain of functions, so to speak (<span><span>V→r→t</span> </span>).

Now what you can do is simply plug in what <span>r </span> is (note you were given diameter) and what <span><span><span>dr</span><span>dt</span></span> </span> is, because <span><span><span>dV</span><span>dt</span></span> </span> describes the rate of change of the volume over time, of a sphere.

<span><span><span><span><span>dV</span><span>dt</span></span>=<span>43</span>π<span>(3<span><span>(20mm)</span>2</span>)</span><span>(4<span>mms</span>)</span></span> </span><span><span>=6400π<span><span>mm3</span>s</span></span> </span></span>

Since time just increases, and the radius increases as a function of time, and the volume increases as a function of a constant times the radius cubed, the volume increases faster than the radius increases, so we can't just say the two rates are the same.

You might be interested in
Pls pls help me out AHH, what is not true about MEIOSIS?
agasfer [191]

Answer:

B.

Explanation:

This is not true as the number of chromosomes in the daughter cells are half the number in the parent's cells

4 0
3 years ago
Cocking your head would be most useful for detecting the ______ of a sound.
statuscvo [17]
Cocking your head would be most useful for detecting the LOCATION of a sound.
5 0
3 years ago
Read 2 more answers
Consider a small frictionless puck perched at the top of a xed sphere of radius R. If the puck is given a tiny nudge so that it
MakcuM [25]

Answer:

Explanation:

Let the vertical height by which it descends be h . Let it acquire velocity of v .

1/2 mv² = mgh

v² = 2gh

As it leaves the surface of sphere , reaction force of surface  R = 0 , so

centripetal force = mg cosθ where θ is the angular displacement from the vertex .  

mv² / r = mg cosθ

(m/r )x 2gh = mg cosθ

2h / r = cosθ

cosθ = (r-h) / r

2h / r =  r-h / r

2h = r-h

3h = r

h = r / 3

5 0
3 years ago
Is a cinder cone volcanoe constructive or deconstructive
Ksivusya [100]
Cinder cone volcanoes can be associated with either constructive or destructive margins.<span>
</span>
6 0
3 years ago
How many atoms are in 3.5 moles of sulfur
irina [24]
There are 2.1077x10^24 atoms of sulfur in 3.50 mols of sulfur.
5 0
2 years ago
Other questions:
  • 1. Use the diagram to anwser question 1.
    6·2 answers
  • Birds resting on high-voltage power lines are a common sight. the copper wire on which a bird stands is 1.28 cm in diameter and
    8·1 answer
  • A force that comes from the action of earth's Gravity is called
    6·2 answers
  • Can someone help me its A,B,C,D
    5·1 answer
  • Light enters an equilateral triangle prism in a direction parallel to one side. The prism is made of glass with an index of refr
    7·1 answer
  • As you approach your vehicle and perform checks it is not necessary to always
    10·1 answer
  • explain y it is easier to loosen a tight but using a spanner with along handle than with a short handle​
    12·1 answer
  • Samples of different materials, A and B, have the same mass, but the sample
    14·1 answer
  • A plant is thrown straight
    15·1 answer
  • In a single-slit diffraction experiment, the width of the slit through which light passes is reduced. what happens to the width
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!