Help me please! If f(x) varies directly with x2 and f(x)=10 when x=−4, what is the value of f(x) when x = 3?

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

Help please……..??.?.?.?.?.?

aleksklad [387]

9514 1404 393

Answer:

c) 16,500 m³

d) 277,088 mm³

a) V = LWH

b) V = πr²h

Step-by-step explanation:

The relevant volume formulas are ...

rectangular pyramid: V = 1/3LWH
cylinder: V = πr²h
rectangular prism: V = LWH

__

13c. The pyramid formula above tells us the volume is ...

V = 1/3(60 m)(15 m)(55 m) = 16,500 m³

__

13d. The cylinder formula above tells us the volume is ...

V = π(35 mm)²(72 mm) ≈ 277,088 mm³ ≈ 277 mL

__

14a. The shape appears to be a rectangular prism, so its volume is given by the formula ...

V = LWH . . . . . where L, W, H represent the length, width, and height

__

14b. The volume of a cylinder is given by the formula ...

V = πr²h . . . . . where r, h represent the radius and height (length)

7 0

2 years ago

Saul wrote Olive a check for $296.45, and Olive deposited the check into her

statuscvo [17]

Answer:

A the front of her check only

Step-by-step explanation:

When you receive a check the back doesn't have anything on the back that you write on hope this helps.

4 0

2 years ago

Would really appreciate the help, can't seem to figure out the answer;-;

Dafna11 [192]

Answer:

-1*70

Step-by-step explanation:

-1*70=-71

7 0

2 years ago

What is the value of y = sin (? when ? = 0°?

Montano1993 [528]

Y=sin(? when ?=0° the answer is 0

Y=0

3 0

2 years ago