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
Goryan [66]
3 years ago
10

(Old Final Exam Problem-Sp2009) An hourglass is made up of two glass cones connected at their tips. Both cones have radius 1 inc

h and height 3 inches. When the hourglass is flipped over, sand starts falling to the lower cone. (a) When the sand remaining in the upper cone has height y inches, its volume A in terms of y is . (b) When the sand in the lower cone has reached a height of h inches, its volume B in terms of h is . (Hint: B is the volume of the bottom cone minus the volume of the empty space above the sand.) (c) Assume the total volume of sand in the hourglass is 3π/4 cubic inches. Also, assume the height of the sand in the upper cone is decreasing at a rate of 4/100 inches per second. At the instant when the sand in the lower cone is 1 inch high, the height of the sand in the lower cone is increasing at a rate of inches per second.
Mathematics
1 answer:
ElenaW [278]3 years ago
7 0

Answer:

  (a)  A = (π/27)y³

  (b)  B = π -(π/27)(3 -h)³

  (c)  0.0116 in/s

Step-by-step explanation:

(a) The volume of a cone is given by the formula ...

  V = 1/3πr²h

For this part of the problem, the volume is designated A and we have h = y and r = y/3, so the volume is ...

  A = (1/3)π(y/3)²(y)

  A = (π/27)y³

__

(b) The empty volume of the lower cone (V) is the same as the volume of the upper cone with y=3, so is ...

  V = (π/27)(3³) = π . . . . . cubic inches

When the sand height is h in the lower cone, the height of the empty space is (3-h), so the volume of the empty space is ...

  empty volume = (π/27)(3-h)³

and the volume B is the difference between that and the cone volume:

  B = cone volume - empty volume

  B = π -(π/27)(3 -h)³

__

(c) We want the rate of change of height in the lower cone at a particular height when the rate of change of height in the upper cone is a particular value. We could sum the volumes and differentiate to find the rate of change of one height with respect to the other. (We would still need to find the height in the upper cone.) Or, we can cut to the chase.

The ratio of rates of change in height will be inversely proportional to the surface areas of the sand in each section of the cone. In turn, that ratio will be proportional to the square of the distance from the sand surface to the cone tip.

When the sand in the lower cone is 1 inch high, its volume is ...

  B(1) = π -(π/27)(3 -1)³ = π -(8π/27) = 19π/27

Then the volume of the sand in the upper cone is ...

  A = 3π/4 -19π/27 = 5π/108

The height of the sand in the upper cone is then ...

  (π/27)h³ = π(5/108)

  h³ = 5/4

  h = ∛(5/4) . . . . . . . height of sand in upper cone

The sand surface in the lower cone is 1 inch above the base, so 2 inches from the apex. The ratio of surface areas in the two cones is then ...

  lower sand area / upper sand area = (2 / ∛(5/4))²

The height of the sand in the lower cone will be increasing at a rate inversely proportional to this and proportional to the rate of decrease of height in the upper cone, given as 4/100 in/s.

  rate of increase of sand in lower cone = (4/100)/((2 / ∛(5/4))²)

  = (∛12.5)/200 ≈ 0.0116 . . . . inches per second

You might be interested in
The size of a television screen is measured by the length of thee diagonal of the screen. What size is a television screen that
Murljashka [212]

Answer:

approximately 6.708 feet

Step-by-step explanation:

We use the Pythagorean theorem to solve the problem, using 3 feet and 6 feet as the legs of a right angle triangle. The diagonal of the screen is therefore the "hypotenuse" of this right angle triangle, and can be determined via the formula:

diagonal = \sqrt{(3\,ft)^2+(6\,ft)^2} \\diagonal = \sqrt{9\,ft^2+36\,ft^2} \\diagonal = \sqrt{45} \,ft\\diagonal = 3\,\sqrt{5} \,ft

which is approximately 6.708 feet

6 0
3 years ago
Graph the function below:<br> y = 32 +1
sveta [45]
Slope: 0 | Y intercept 0, 33
7 0
3 years ago
Which of the following is a monomial?
Semmy [17]

Answer:

C

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Find the least common multiple of 7,9 and 21
lawyer [7]
Free LCM Calculator determines the least common multiple (LCM) between 9 and 21 the smallest integer that is 63 that is divisible by both numbers. Least Common Multiple (LCM) of 9 and 21 is 63.
...
Prime Factorization of 21.
3 21
7 7
1
3 0
3 years ago
Simplify the following expression:<br> 4x - 3y + 2z - 3x - y + 3z = ?
nordsb [41]
Add all the like terms together so 4x-3x is x or 1x -3y-y is -4y 2z+3z is 5z so put them all together it's x-4y+5z
7 0
3 years ago
Other questions:
  • It takes 3/4 yards of ribbon to make a bow how many bows can be made with 7 yards
    5·2 answers
  • the chef has eight blocks of butter. she cuts the butter into fourths. how many slices of butter does the chef have now​
    6·2 answers
  • Michelle likes chocolate but it causes acne. She chooses to limit herself to three pieces of chocolate every week. Except the fi
    13·2 answers
  • How do you find surface area of a cylinder in simple words/terms?
    6·2 answers
  • Mrs roscoe can bake A batch of 8 cup cakes in 3/4 hours. how many cupcakes can she bake and 5 1/4 hours
    9·2 answers
  • Explain the process for finding the product of two integers
    6·1 answer
  • Help with math please
    11·1 answer
  • Will factor the express 36xyz + 24xy − 16x by dividing each term by a common fact
    5·2 answers
  • What is the slope and the y-intercept.
    11·1 answer
  • Whats the answer pls i need help
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!