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3 years ago

If x + y = 5 and at the same time 2x - y = 7. find the value of x

Mathematics

1 answer:

Fed [463]3 years ago

5 0

Answer:

x = 2y − 2

Step-by-step explanation:

x + y + 5 = 2x - y + 7

−x + y + 5 = −y + 7

−x + 5 = −2y +7

−x = −2y + 2

x = 2y − 2

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The answer i think is it c

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3 years ago

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What are the real and complex solutions of the polynomial equation? x^3-8=0. with imaginary numbers

seraphim [82]

Answer:

Solutions are 2, -1 + 0.5 sqrt10 i and -1 - 0.5 sqrt10 i

or 2, -1 + 1.58 i and -1 - 1.58i

(where the last 2 are equal to nearest hundredth).

Step-by-step explanation:

The real solution is x = 2:-

x^3 - 8 = 0

x^3 = 8

x = cube root of 8 = 2

Note that a cubic equation must have a total of 3 roots ( real and complex in this case). We can find the 2 complex roots by using the following identity:-

a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Here a = x and b = 2 so we have

(x - 2)(x^2 + 2x + 4) = 0

To find the complex roots we solve x^2 + 2x + 4 = 0:-

Using the quadratic formula x = [-2 +/- sqrt(2^2 - 4*1*4)] / 2

= -1 +/- (sqrt( -10)) / 2

= -1 + 0.5 sqrt10 i and -1 - 0.5 sqrt10 i

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3 years ago

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The surface area of a right circular cone of radius r and height h is S = πr√ r 2 + h 2 , and its volume is V = 1 3 πr2h. What i

kirill115 [55]

Answer:

Required largest volume is 0.407114 unit.

Step-by-step explanation:

Given surface area of a right circular cone of radious r and height h is,

$S=\pi r\sqrt{r^2+h^2}$

and volume,

$V=\frac{1}{3}\pi r^2 h$

To find the largest volume if the surface area is S=8 (say), then applying Lagranges multipliers,

$f(r,h)=\frac{1}{3}\pi r^2 h$

subject to,

$g(r,h)=\pi r\sqrt{r^2+h^2}=8\hfill (1)$

We know for maximum volume $r\neq 0$ . So let $\lambda$ be the Lagranges multipliers be such that,

$f_r=\lambda g_r$

$\implies \frac{2}{3}\pi r h=\lambda (\pi \sqrt{r^2+h^2}+\frac{\pi r^2}{\sqrt{r^2+h^2}})$

$\implies \frac{2}{3}r h= \lambda (\sqrt{r^2+h^2}+\frac{ r^2}{\sqrt{r^2+h^2}})\hfill (2)$

And,

$f_h=\lambda g_h$

$\implies \frac{1}{3}\pi r^2=\lambda \frac{\pi rh}{\sqrt{r^2+h^2}}$

$\implies \lambda=\frac{r\sqrt{r^2+h^2}}{3h}\hfill (3)$

Substitute (3) in (2) we get,

$\frac{2}{3}rh=\frac{r\sqrt{R^2+h^2}}{3h}(\sqrt{R^2+h^2+}+\frac{r^2}{\sqrt{r^2+h^2}})$

$\implies \frac{2}{3}rh=\frac{r}{3h}(2r^2+h^2)$

$\implies h^2=2r^2$

Substitute this value in (1) we get,

$\pi r\sqrt{h^2+r^2}=8$

$\implies \pi r \sqrt{2r^2+r^2}=8$

$\implies r=\sqrt{\frac{8}{\pi\sqrt{3}}}\equiv 1.21252$

Then,

$h=\sqrt{2}(1.21252)\equiv 1.71476$

Hence largest volume,

$V=\frac{1}{3}\times \pi \times\frac{\pi}{8\sqrt{3}}\times 1.71476=0.407114$

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3 years ago

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damaskus [11]

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4 years ago

A 22-ft ladder leans against a building so that the angle between the ground and the ladder is 64°. How high does the ladder rea

vodka [1.7K]

The height of the ladder on the building is 19.77 feet

<h3>How high does the ladder reach on the building?</h3>

Represent the height of the ladder on the building with h

So, the given parameters are:

Angle, x = 64 degrees

Length of ladder, l = 22 feet

The height of the ladder on the building is calculated using

sin(x) = h/l

Substitute the known values in the above equation

sin(64) = h/22

Multiply both sides by 22

h = 22 * sin(64)

Evaluate the product

h = 19.77

Hence, the height of the ladder on the building is 19.77 feet