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3 years ago

A circle passes through (3,-1),(-2, 4), and (6,8)

Mathematics

1 answer:

Vaselesa [24]3 years ago

5 0

Answer:

D = -6, E = -8 , F = 0

Step-by-step explanation:

standard form = $x^{2} + y^{2} + Dx + Ey + F = 0$

now, when circle passes through (3,-1);

⇒ $3^{2} + (-1)^{2} + D(3) + E(-1) + F = 0$

⇒ $3D - E + F + 10 = 0$ ...............( equation 1)

when circle passes through (-2,4);

⇒ $(-2)^{2} + 4^{2} + D(-2) + E(4)+ F = 0$

⇒ $-2D + 4E + F + 20 = 0$ ...............( equation 2)

when circle passes through (6,8);

⇒ $6^{2} + 8^{2} + D(6) + E(8) + F = 0$

⇒ $6D + 8E + F + 100= 0$ ................( equation 3)

by solving these 3 equations , we get;

D = -6, E = -8, F = 0

hence,

standard form = $x^{2} + y^{2} + Dx + Ey + F = 0$

= $x^{2} + y^{2} - 6x - 8y = 0$

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Given P = x^0.3 y^0.7 is the chicken lay eggs production function, where P is the number of eggs lay, x is the number of workers

lora16 [44]

Answer:

Part A)

$\displaystyle \frac{dy}{dx}=-\frac{3}{7}P^\frac{10}{7}x^{-\frac{10}{7}}$

Part B)

The daily operating cost decreases by about $143 per extra worker.

Step-by-step explanation:

We are given the equation:

$\displaystyle P=x^{\frac{3}{10}}y^{\frac{7}{10}}$

Where P is the number of eggs laid, x is the number of workers, and y is the daily operating budget (assuming in US dollars $).

We want to find dy/dx.

So, let’s find our equation in terms of x. We can raise both sides to 10/7. Hence:

$\displaystyle P^\frac{10}{7}=\Big(x^\frac{3}{10}y^\frac{7}{10}\Big)^\frac{10}{7}$

Simplify:

$\displaystyle P^\frac{10}{7}=x^\frac{3}{7}y$

Divide both sides by the x term to acquire:

$\displaystyle y=P^\frac{10}{7}x^{-\frac{3}{7}}$

Take the derivative of both sides with respect to x:

$\displaystyle \frac{dy}{dx}=\frac{d}{dx}\Big[P^\frac{10}{7}x^{-\frac{3}{7}}\Big]$

Apply power rule. Note that P is simply a constant. Hence:

$\displaystyle \frac{dy}{dx}=P^\frac{10}{7}(-\frac{3}{7})(x^{-\frac{10}{7}})$

Simplify. Hence, our derivative is:

$\displaystyle \frac{dy}{dx}=-\frac{3}{7}P^\frac{10}{7}x^{-\frac{10}{7}}$

Part B)

We want to evaluate the derivative when x is 30 and when y is $10,000.

First, we will need to find P. Our original equations tells us that:

$P=x^{0.3}y^{0.7}$

Hence, at x = 30 and at y = 10,000, P is:

$P=(30)^{0.3}(10000)^{0.7}$

Therefore, for our derivative, we will have:

$\displaystyle \frac{dy}{dx}=-\frac{3}{7}\Big(30^{0.3}(10000^{0.7})\Big)^\frac{10}{7}\Big(30^{-\frac{10}{7}}\Big)$

Use a calculator. So:

$\displaystyle \frac{dy}{dx}=-\frac{1000}{7}=-142.857142...\approx-143$

Our derivative is given by dy/dx. So, it represents the change in the daily operating cost over the change in the number of workers.

So, when there are 30 workers with a daily operating cost of $10,000 producing a total of about 1750 eggs, the daily operating cost decreases by about $143 per extra worker.

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2 years ago

Please help me out on this question

Mariulka [41]

At x=2, y=4

At x=4, y=16

So the total change in y from x=2 to x=4 is 16-4=12

And since average rate of change = total change in y divided by total change in x within the same period, therefore the answer is 12/2 = 6

Hope that helps. Let me know if you have any questions.

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Step-by-step explanation:

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