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3 years ago

Value of 3x^2 +4y^2 if x=2, y=1, and z= -3

Mathematics

1 answer:

Dennis_Churaev [7]3 years ago

4 0

$3x^2+4y^2\\\\\text{Put the values of x=2, y=1 and z=-3 to the expression:}\\\\3(2^2)+4(1^2)=3(4)+4(1)=12+4=16$

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Find all solutions of each equation on the interval 0 ≤ x < 2π.

Korvikt [17]

Answer:

$x = 0$ or $x = \pi$ .

Step-by-step explanation:

How are tangents and secants related to sines and cosines?

$\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}$ .

$\displaystyle \sec{x} = \frac{1}{\cos{x}}$ .

Sticking to either cosine or sine might help simplify the calculation. By the Pythagorean Theorem, $\sin^{2}{x} = 1 - \cos^{2}{x}$ . Therefore, for the square of tangents,

$\displaystyle \tan^{2}{x} = \frac{\sin^{2}{x}}{\cos^{2}{x}} = \frac{1 - \cos^{2}{x}}{\cos^{2}{x}}$ .

This equation will thus become:

$\displaystyle \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} \cdot \frac{1}{\cos^{2}{x}} + \frac{2}{\cos^{2}{x}} - \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} = 2$ .

To simplify the calculations, replace all $\cos^{2}{x}$ with another variable. For example, let $u = \cos^{2}{x}$ . Keep in mind that $0 \le \cos^{2}{x} \le 1 \implies 0 \le u \le 1$ .

$\displaystyle \frac{1 - u}{u^{2}} + \frac{2}{u} - \frac{1 - u}{u} = 2$ .

$\displaystyle \frac{(1 - u) + u - u \cdot (1- u)}{u^{2}} = 2$ .

Solve this equation for $u$ :

$\displaystyle \frac{u^{2} + 1}{u^{2}} = 2$ .

$u^{2} + 1 = 2 u^{2}$ .

$u^{2} = 1$ .

Given that $0 \le u \le 1$ , $u = 1$ is the only possible solution.

$\cos^{2}{x} = 1$ ,

$x = k \pi$ , where $k\in \mathbb{Z}$ (i.e., $k$ is an integer.)

Given that $0 \le x < 2\pi$ ,

$0 \le k$ .

$k = 0$ or $k = 1$ . Accordingly,

$x = 0$ or $x = \pi$ .

8 0

2 years ago

Read 2 more answers

A right rectangular prism has edges of 2 1/4 in, 1 1/2 in and 1 1/2 in. How many cubes with lengths of 1/4 in would be needed to

Zielflug [23.3K]

The volume of a rectangular prism is (length) x (width) x (height).

The volume of the big one is (2.25) x (1.5) x (1.5) = 5.0625 cubic inches.

The volume of the little one is (0.25)x(0.25)x(0.25)= 0.015625 cubic inch

The number of little ones needed to fill the big one is

 (Volume of the big one) divided by (volume of the little one) .

 5.0625 / 0.015625 = 324 tiny cubies

=================================================

Doing it with fractions instead of decimals:

The volume of a rectangular prism is (length) x (width) x (height).
Dimensions of the big one are:

 2-1/4 = 9/4
 1-1/2 = 3/2
 1-1/2 = 3/2

 Volume = (9/4) x (3/2) x (3/2) =

 (9 x 3 x 3) / (4 x 2 x 2) =

 81 / 16 cubic inches.

As a mixed number: 81/16 = 5-1/16 cubic inches

Volume of the tiny cubie = (1/4) x (1/4) x (1/4) = 1/64 cubic inch.

The number of little ones needed to fill the big one is

 (Volume of the big one) divided by (volume of the little one) .

 (81/16) divided by (1/64) =

 (81/16) times (64/1) =

 5,184/16 = 324 tiny cubies.

5 0

3 years ago

5th grade . pls help.

fiasKO [112]

4/16 and 1/4 happy to help :)

8 0

2 years ago

The graph shows a 90° rotation. The matrix M such that, for any point (x, y), M(x, y) = (x', y') is

guapka [62]

Answer:

A) $\left[\begin{array}{cc}0&-1\\1&0\end{array}\right]$