Help Pa help po sa differential calculus

[matt43]

Convert the Cartesian coordinate (9, -9,2) into cylindrical coordinates.
A. (9 sq. ft. of 2, 7n/4, 2)
B. (9 sq. rt. Of 2, 5n/4, 2)
C. (-9 sq. rt. Of 2, 3n/4, 2)
D. (-9 sq. ft. of 2, 7n/4, 2)

 

[🤖]

To convert from cartesian coordinates $(x,y,z)$ to cylindrical coordinates $(r,\theta,z)$, we use the following formulas:

$$r=\sqrt{x^2+y^2}, \quad \theta = \tan^{-1}(y/x), \quad z=z$$

Applying these formulas to the point $(9,-9,2)$, we get:

$$r=\sqrt{9^2+(-9)^2}=9\sqrt{2}$$

$$\theta=\tan^{-1}(-9/9)=-\frac{\pi}{4}+\pi n, \quad \text{where } n \in \mathbb{Z}$$

Note that we get a negative angle because the point is in the third quadrant. We can add any integer multiple of $\pi$ to get all possible values of $\theta$ for this point. Therefore, we can write:

$$\theta=-\frac{\pi}{4}+\pi n, \quad \text{where } n \in \mathbb{Z}$$

Finally, we have $z=2$. So the cylindrical coordinates of the point $(9,-9,2)$ are:

$$(r,\theta,z)=\boxed{(9\sqrt{2}, -\frac{\pi}{4}+\pi n, 2)}, \quad \text{where } n \in \mathbb{Z}$$

Therefore, the answer is none of the choices given.

 

[matt43]

A rectangular box with square base and open at the top is to have a capacity of 16823 cu.cm. Find the height of the box that requires minimum amount of material required.

 

[🤖]

Let the side length of the square base be x and let the height of the box be h. Then the volume of the box is given by:

V = x^2 h = 16823

We want to minimize the amount of material required, which is proportional to the surface area of the box. The surface area includes the area of the square base, which is x^2, and the area of the four rectangular sides, each of which has area xh. Thus, the total surface area is:

A = x^2 + 4xh

Using the volume equation to eliminate h, we get:

h = 16823/x^2

Substituting this into the expression for surface area, we get:

A = x^2 + 4x(16823/x^2) = x^2 + 67292/x

To minimize A, we take the derivative with respect to x and set it equal to zero:

dA/dx = 2x - 67292/x^2 = 0

Solving for x, we get:

x = sqrt(67292/2) = 146.05 cm (to two decimal places)

Substituting this value of x back into the volume equation, we get:

h = 16823/146.05^2 = 8.59 cm (to two decimal places)

Therefore, the height of the box that requires minimum amount of material is approximately 8.59 cm.

 

[matt43]

A rectangular box with square base and open at the top is to have a capacity of 16823 cu.cm. Find the height of the box that requires minimum amount of material required.
a. 16.14 cm
b. 14.12 cm
c. 12.14 cm
d. 10.36 cm