A number is 4 hundreds more than 142. What is the number?

Roman55 [17]

The two parabolas intersect for

$8-x^2 = x^2 \implies 2x^2 = 8 \implies x^2 = 4 \implies x=\pm2$

and so the base of each solid is the set

$B = \left\{(x,y) \,:\, -2\le x\le2 \text{ and } x^2 \le y \le 8-x^2\right\}$

The side length of each cross section that coincides with B is equal to the vertical distance between the two parabolas, $|x^2-(8-x^2)| = 2|x^2-4|$ . But since -2 ≤ x ≤ 2, this reduces to $2(x^2-4)$ .

a. Square cross sections will contribute a volume of

$\left(2(x^2-4)\right)^2 \, \Delta x = 4(x^2-4)^2 \, \Delta x$

where ∆x is the thickness of the section. Then the volume would be

$\displaystyle \int_{-2}^2 4(x^2-4)^2 \, dx = 8 \int_0^2 (x^2-4)^2 \, dx \\\\ = 8 \int_0^2 (x^4-8x^2+16) \, dx \\\\ = 8 \left(\frac{2^5}5 - \frac{8\times2^3}3 + 16\times2\right) = \boxed{\frac{2048}{15}}$

where we take advantage of symmetry in the first line.

b. For a semicircle, the side length we found earlier corresponds to diameter. Each semicircular cross section will contribute a volume of

$\dfrac\pi8 \left(2(x^2-4)\right)^2 \, \Delta x = \dfrac\pi2 (x^2-4)^2 \, \Delta x$

We end up with the same integral as before except for the leading constant:

$\displaystyle \int_{-2}^2 \frac\pi2 (x^2-4)^2 \, dx = \pi \int_0^2 (x^2-4)^2 \, dx$

Using the result of part (a), the volume is

$\displaystyle \frac\pi8 \times 8 \int_0^2 (x^2-4)^2 \, dx = \boxed{\frac{256\pi}{15}}}$

c. An equilateral triangle with side length s has area √3/4 s², hence the volume of a given section is

$\dfrac{\sqrt3}4 \left(2(x^2-4)\right)^2 \, \Delta x = \sqrt3 (x^2-4)^2 \, \Delta x$

and using the result of part (a) again, the volume is

$\displaystyle \int_{-2}^2 \sqrt 3(x^2-4)^2 \, dx = \frac{\sqrt3}4 \times 8 \int_0^2 (x^2-4)^2 \, dx = \boxed{\frac{512}{5\sqrt3}}$

7 0

2 years ago

What is the twentieth term of the arithmetic sequence described by the<br> equation an= 3n - 10?

sergij07 [2.7K]

Answer:

The twentieth term is 50.

Step-by-step explanation:

The equation that describes the arithmetic sequence describes it's "general term", which means that all the numbers in that sequence must follow that equation. For instance, if we want to find the first term of the sequence we must make a = 1 and we have:

a1 = 3*1 - 10

a1 = -7

Therefore to find the twentieth term we need to make n equal to 20. This is done below:

a20 = 3*20 - 10

a20 = 60 - 10

a20 = 50

The twentieth term is 50.

8 0

3 years ago

Write the prime factorization of the number 144 243 475

postnew [5]

18. There are Personal Muth Trainer library. The 318 fiction books in the class ne number number of nonfiction books is 47 less than of fiction books. Part A About how many nonfiction books are there in the class library? Explain. Part B How many fiction and nonfiction books are there in the class library altogether? Show your work.

4 0

3 years ago

The line of reflection is blank and the coordinates of C' are blank ?

denpristay [2]

Answer:

I did this before it is C

Step-by-step explanation:

4 0

2 years ago

What is the value of x?<br><br> 8x−10−4x=−12(12x−20)

madreJ [45]

What is the value of X?

Simplify:
8x - 10 - 4x = - 12(12x - 20)

Simplify both sides:
8x - 10 - 4x = - 12(12x -20)

8x + - 10 + - 4x = (- 12)(12x) + ( -12)(- 20)
(Distribute:)
8x + - 10 + - 4x = 144x 240

(8x + - 4x) + ( -10) = - 144x + 240 (Combine Like Terms:)

4x + - 10 = - 144x + 240

4x - 10 = - 144x + 240

Add - 144x to both sides:

4x - 10 + 144x = - 144x + 240 + 144x

148x - 10 = 240

Add 10 to both sides:

148x - 10 + 10 = 240 + 10

148x = 250

Divide both sides by 148

148x/148 = 250/148

x = 125/74

Answer: x = 125/74 (Decimal: 1.689189)

Hope that helps!!!!

3 0

3 years ago