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emmasim [6.3K]
3 years ago
7

Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (−1, 0).

Mathematics
1 answer:
Hitman42 [59]3 years ago
7 0
<span>The two points that are most distant from (-1,0) are exactly (1/3, 4sqrt(2)/3) and (1/3, -4sqrt(2)/3) approximately (0.3333333, 1.885618) and (0.3333333, -1.885618) Rewriting to express Y as a function of X, we get 4x^2 + y^2 = 4 y^2 = 4 - 4x^2 y = +/- sqrt(4 - 4x^2) So that indicates that the range of values for X is -1 to 1. Also the range of values for Y is from -2 to 2. Additionally, the ellipse is centered upon the origin and is symmetrical to both the X and Y axis. So let's just look at the positive Y values and upon finding the maximum distance, simply reflect that point across the X axis. So y = sqrt(4-4x^2) distance is sqrt((x + 1)^2 + sqrt(4-4x^2)^2) =sqrt(x^2 + 2x + 1 + 4 - 4x^2) =sqrt(-3x^2 + 2x + 5) And to simplify things, the maximum distance will also have the maximum squared distance, so square the equation, giving -3x^2 + 2x + 5 Now the maximum will happen where the first derivative is equal to 0, so calculate the first derivative. d = -3x^2 + 2x + 5 d' = -6x + 2 And set d' to 0 and solve for x, so 0 = -6x + 2 -2 = -6x 1/3 = x So the furthest point will be where X = 1/3. Calculate those points using (1) above. y = +/- sqrt(4 - 4x^2) y = +/- sqrt(4 - 4(1/3)^2) y = +/- sqrt(4 - 4(1/9)) y = +/- sqrt(4 - 4/9) y = +/- sqrt(3 5/9) y = +/- sqrt(32)/sqrt(9) y = +/- 4sqrt(2)/3 y is approximately +/- 1.885618</span>
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n your computer store you charge $1012 per computer sold. Your costs are given by the equation C(x) = 4x^2 + 4x - 80 where x is
ella [17]

Answer:

63568 dollars

Step-by-step explanation:

Given that in your computer store you charge $1012 per computer sold. Your costs are given by the equation

C(x) = 4x^2 + 4x - 80

where x is the number of computers sold.

Revenue = sales price *no of computers sold

= 1012x

Profit = Revenue - cost

= 1012x-( 4x^2 + 4x - 80)\\= 1008x-4x^2+80\\

Use derivative test to find maximum profit

C'(x) = 1008-8x \\C''(x)= -x

Equate I derivative to 0

x= 128

i.e. if 128 computers are manufactured and sold profit wouldbemaximum

Profit maximum=1008(128)-4(128)^2+80\\=63568

3 0
3 years ago
Volume and surface area are often compared by manufacturers in order to maximize how much of something can go inside of a packag
notka56 [123]
We have no dimensions to work with. I'll pick some and try and comply with the conditions of the problem. 

Suppose you have an object that is 14 by 22 by 27 cm. These three numbers have no common factor so they cannot be reduced any further, which is helpful for this problem.

Find the Volume
Volume

l = 27 cm
w = 14 cm
h = 22 cm

V = 27 *14 * 22
V = 8316 cm^3

Find the surface area
SA = 2*l*w + 2*l*h  + 2*w*h
SA = 2*27*14  +  2*27*22   +  2*14*22
SA = 756 + 1188 + 616
SA = 2558

Just looking at these numbers The surface area is about 1/3 of the volume. I don't think this is always true.


 \frac{SA}{V} =  \frac{2*L*W  + 2*L*H + 2*W*H}{L*W*H} 

Another way to do this is to consider a cube which might give you a more useful result. 

s = L = W = H all three dimensions are equal in a cube.
The volume of a cube is s*s*s = s^3
The surface area of a cube is 2*s*s + 2*s*s + 2s*s = 6s^2

\frac{SA_cube}{V_cube} =  \frac{6s*s}{s*s*s}
\frac{SA}{V} =  \frac{6}{s}

That means whatever the side length, the Surface Area to volume = 6/the side length which is kind of an interesting result.


4 0
3 years ago
Could you please solve that?
nordsb [41]

Answer:

i think my brain just died what math are you doing

Step-by-step explanation:

3 0
3 years ago
F(x) = 2(x - 3)^2 - 2
Natali5045456 [20]

Hi. I was unsure of what exactly you wanted from this equation, so here's a quick analysis:

<em>f(x) = 2(x - 3)^2 - 2</em>

<em></em>

Domain: (-∞, ∞)

Range: (-2, ∞)

X-intercepts: (4, 0), (2, 0)

Y-intercept: (0, 16)

Axis of Symmetry: x = 3

Minimum value (vertex): (3, -2)

Standard form: y = 2x^2 - 12x + 16

4 0
2 years ago
Graph the following function on the axes provided.<br>​
Bingel [31]

Answer:

Step-by-step explanation:

6 0
2 years ago
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