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
kompoz [17]
3 years ago
5

use the general slicing method to find the volume of The solid whose base is the triangle with vertices (0 comma 0 )​, (15 comma

0 )​, and (0 comma 15 )and whose cross sections perpendicular to the base and parallel to the​ y-axis are semicircles

Mathematics
1 answer:
lyudmila [28]3 years ago
3 0

Answer:

volume V of the solid

\boxed{V=\displaystyle\frac{125\pi}{12}}

Step-by-step explanation:

The situation is depicted in the picture attached

(see picture)

First, we divide the segment [0, 5] on the X-axis into n equal parts of length 5/n each

[0, 5/n], [5/n, 2(5/n)], [2(5/n), 3(5/n)],..., [(n-1)(5/n), 5]

Now, we slice our solid into n slices.  

Each slice is a quarter of cylinder 5/n thick and has a radius of  

-k(5/n) + 5  for each k = 1,2,..., n (see picture)

So the volume of each slice is  

\displaystyle\frac{\pi(-k(5/n) + 5 )^2*(5/n)}{4}

for k=1,2,..., n

We then add up the volumes of all these slices

\displaystyle\frac{\pi(-(5/n) + 5 )^2*(5/n)}{4}+\displaystyle\frac{\pi(-2(5/n) + 5 )^2*(5/n)}{4}+...+\displaystyle\frac{\pi(-n(5/n) + 5 )^2*(5/n)}{4}

Notice that the last term of the sum vanishes. After making up the expression a little, we get

\displaystyle\frac{5\pi}{4n}\left[(-(5/n)+5)^2+(-2(5/n)+5)^2+...+(-(n-1)(5/n)+5)^2\right]=\\\\\displaystyle\frac{5\pi}{4n}\displaystyle\sum_{k=1}^{n-1}(-k(5/n)+5)^2

But

\displaystyle\frac{5\pi}{4n}\displaystyle\sum_{k=1}^{n-1}(-k(5/n)+5)^2=\displaystyle\frac{5\pi}{4n}\displaystyle\sum_{k=1}^{n-1}((5/n)^2k^2-(50/n)k+25)=\\\\\displaystyle\frac{5\pi}{4n}\left((5/n)^2\displaystyle\sum_{k=1}^{n-1}k^2-(50/n)\displaystyle\sum_{k=1}^{n-1}k+25(n-1)\right)

we also know that

\displaystyle\sum_{k=1}^{n-1}k^2=\displaystyle\frac{n(n-1)(2n-1)}{6}

and

\displaystyle\sum_{k=1}^{n-1}k=\displaystyle\frac{n(n-1)}{2}

so we have, after replacing and simplifying, the sum of the slices equals

\displaystyle\frac{5\pi}{4n}\left((5/n)^2\displaystyle\sum_{k=1}^{n-1}k^2-(50/n)\displaystyle\sum_{k=1}^{n-1}k+25(n-1)\right)=\\\\=\displaystyle\frac{5\pi}{4n}\left(\displaystyle\frac{25}{n^2}.\displaystyle\frac{n(n-1)(2n-1)}{6}-\displaystyle\frac{50}{n}.\displaystyle\frac{n(n-1)}{2}+25(n-1)\right)=\\\\=\displaystyle\frac{125\pi}{24}.\displaystyle\frac{n(n-1)(2n-1)}{n^3}

Now we take the limit when n tends to infinite (the slices get thinner and thinner)

\displaystyle\frac{125\pi}{24}\displaystyle\lim_{n \rightarrow \infty}\displaystyle\frac{n(n-1)(2n-1)}{n^3}=\displaystyle\frac{125\pi}{24}\displaystyle\lim_{n \rightarrow \infty}(2-3/n+1/n^2)=\\\\=\displaystyle\frac{125\pi}{24}.2=\displaystyle\frac{125\pi}{12}

and the volume V of our solid is

\boxed{V=\displaystyle\frac{125\pi}{12}}

You might be interested in
Tuto
Damm [24]

Answer:

8r+8pg-pq

Step-by-step explanation:

The subtractable pg cancels out one of the 9 pg's. So 9 pg-1 pg= 8 pg

Hope this helps!

8 0
3 years ago
Let f(x) = x2 + x + 2 and g(x) = 2x2 + 5. Find f(g(x)). Show each step of your work.
Tju [1.3M]
G(x)=9
f(g(x))= 9(2)+9+2=29
5 0
2 years ago
Five members of the soccer team ordered orange juice. Ten members ordered apple juice. Juice was packaged in 250 mL cartons. Exp
Lady_Fox [76]
The answer to the question is B
6 0
2 years ago
Given the data, as shown in the image below, determine if the distribution is uniformly distributed, symmetrically distributed,
Sedbober [7]

Answer:

The distribution is symmetric.

Step-by-step explanation:

<em>The distribution will be skew left, if the data is more distributed on left side of graph.</em>

<em>The distribution will be right left, if the data is more distributed on right side of graph.</em>

<em>The distribution is symmetric if from the center, the data is distributed symmetrically, equal increase or decrease on either side.</em>

<em>The distribution is uniform if the value of data remains constant throughout the graph.</em>

Above here, the from the center, the data decreases symmetrically on both the sides, same values of data for Cat-Rabbit pair and Dog-Mice pair.

Thus, distribution is symmetric.

8 0
2 years ago
Evaluate when x = -3 and y= -1:<br><br><br> x-xy=
LenKa [72]
-3-(-3)(-1)= 0, so 0 is the solution
5 0
3 years ago
Read 2 more answers
Other questions:
  • Given the Venn diagram below, which statement must be false?
    13·1 answer
  • Maya has 120 caramel apples to sell. Each caramel apple is covered with one topping. 1/5 of the caramel apples are covered with
    8·2 answers
  • helppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
    11·1 answer
  • A toaster oven usually sells for $64. If the toaster oven is 30% off, and sales tax is 5%, what is the total price of the toaste
    12·1 answer
  • 15 points please help thank you
    13·1 answer
  • 7 4/6+1/3−5/6= <br> in fraction thank you
    10·2 answers
  • Pleaseeee helpppp !!!!!!!! Will mark Brianliest !!!!!!!!!!!!!
    7·2 answers
  • 1) What operation is represented in the expression 4(x+4)?
    8·2 answers
  • How to solve 3x+y=6 and y=-3 using substation method
    8·1 answer
  • Solve for x!!!!!!!!! Spam will be taken down
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!