# Univ. Calculus: vol. of spherical raindrop, temp & pressure in fixed vol of gas, ...

#### Onur ERKAYA

##### New member
Univ. Calculus: vol. of spherical raindrop, temp & pressure in fixed vol of gas, ...

Hi everyone,

I need help for solve these problems.

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41. The volume of a falling spherical raindrop grows at a rate which is proportional to the surface area of the drop. Show that the radius of the drop increases at a constant rate.

42. The temperature of the atmosphere decreases with altitude at a rate of 2[SUP]o[/SUP] C per kilometer at the top of a certain cliff. A hang-glider pilot finds that the outside temperature is rising at the rate of (10[SUP]-4[/SUP])[SUP]o[/SUP] C per second. How fast is the glider falling?

43. For temperatures in the range [-50, 150] (degrees Celsius), the pressure in a certain closed container of gas changes linearly with the temperature. Suppose that the 40[SUP]o[/SUP] increase in temperature causes the pressure to increase by 30 millibars. (A millibar is one-thousandth of the average atmospheric pressure at sea level.)
(a) What is the rate of change of pressure with respect to temperature?

(b) What change of temperature would cause the pressure to drop by 9 millibars?

44. Find the rate of change of the length of an edge of a cube with respect to its surface area.

45. The organism amoebus rectilineus always maintains the shape of a right triangle whose area is 10[SUP]-6[/SUP] square millimeters. Find the rate of change of the perimeter at a moment when the organism is isosceles and one of the legs is growing at 10[SUP]-4[/SUP] millimeters per second.

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I tried to solve it but i did not succeed it. Maybe you can help me?

Thank you!

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#### ksdhart2

##### Full Member
I need help for solve these problems.

[HR][/HR]
41. The volume of a falling spherical raindrop grows at a rate which is proportional to the surface area of the drop. Show that the radius of the drop increases at a constant rate.

42. The temperature of the atmosphere decreases with altitude at a rate of 2[SUP]o[/SUP] C per kilometer at the top of a certain cliff. A hang-glider pilot finds that the outside temperature is rising at the rate of (10[SUP]-4[/SUP])[SUP]o[/SUP] C per second. How fast is the glider falling?

43. For temperatures in the range [-50, 150] (degrees Celsius), the pressure in a certain closed container of gas changes linearly with the temperature. Suppose that the 40[SUP]o[/SUP] increase in temperature causes the pressure to increase by 30 millibars. (A millibar is one-thousandth of the average atmospheric pressure at sea level.)
(a) What is the rate of change of pressure with respect to temperature?

(b) What change of temperature would cause the pressure to drop by 9 millibars?

44. Find the rate of change of the length of an edge of a cube with respect to its surface area.

45. The organism amoebus rectilineus always maintains the shape of a right triangle whose area is 10[SUP]-6[/SUP] square millimeters. Find the rate of change of the perimeter at a moment when the organism is isosceles and one of the legs is growing at 10[SUP]-4[/SUP] millimeters per second.

[HR][/HR]
I tried to solve it but i did not succeed it. Maybe you can help me?
You say you've tried these problems. That's great! That means you ought to have loads of work to share with us, as per the rules in the Read Before Posting thread that's stickied at the top of each sub-forum (you did read it, right? ). Please show all of the work you've done on these problems, even the parts you know for sure are wrong. Thank you.

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#### stapel

##### Super Moderator
Staff member
I need help for solve these problems.
What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Also, is it correct to gather that you are now studying "Rates of Change and the Chain Rule", and that these questions are on page 134 (in Chapter 2) of Marsden and Weinstein's "Calculus I", published in the 1980's?

41. The volume of a falling spherical raindrop grows at a rate which is proportional to the surface area of the drop. Show that the radius of the drop increases at a constant rate.
What is the formula for the volume V of a sphere with radius r? What is the formula for the surface area SA of a sphere with radius r? You are given that dV/dt is proportional to SA. What equation does this give you? Where does this lead?

42. The temperature of the atmosphere decreases with altitude at a rate of 2[SUP]o[/SUP] C per kilometer at the top of a certain cliff. A hang-glider pilot finds that the outside temperature is rising at the rate of (10[SUP]-4[/SUP])[SUP]o[/SUP] C per second. How fast is the glider falling?
You are given that the temperature T decreases with the altitude (or height) h, so dT/dh = -2. You are given that the pilot is measuring the temperature change with respect to time of dT/dt = 0.0001. You are asked to find dh/dt. How can you use the givens to create an equation which allows you to find the requested information? Where did this lead? (I think the answer should be in terms of a small number of centimeters per second.)

43. For temperatures in the range [-50, 150] (degrees Celsius), the pressure in a certain closed container of gas changes linearly with the temperature. Suppose that the 40[SUP]o[/SUP] increase in temperature causes the pressure to increase by 30 millibars. (A millibar is one-thousandth of the average atmospheric pressure at sea level.)
(a) What is the rate of change of pressure with respect to temperature?

(b) What change of temperature would cause the pressure to drop by 9 millibars?
What sort of equation does a "linear" relationship have? If the initial temperature was T [SUB]0[/SUB] and the temperature after the increase was T[SUB]1[/SUB], what can you do with the relationship? Where does this lead?

44. Find the rate of change of the length of an edge of a cube with respect to its surface area.
What is the formula for the surface area SA of a cube with side-length s? What then is the formula for the side-length s in terms of the surface area SA? Where does this lead?

45. The organism amoebus rectilineus always maintains the shape of a right triangle whose area is 10[SUP]-6[/SUP] square millimeters. Find the rate of change of the perimeter at a moment when the organism is isosceles and one of the legs is growing at 10[SUP]-4[/SUP] millimeters per second.
What relationship always holds true for the two legs ("a" and "b) and the hypotenuse ("c") of a right triangle? What will be the formula for the perimeter P of this triangle? What is the formula for the area A of this triangle? What then is the length of each of the legs at the moment of evaluation? Where did this lead?