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1 year ago

which of the following expresses the coordinates of the foci of the conic section shown below? (x-2)^2/4+(y+5)^2/9

Mathematics

1 answer:

elena55 [62]1 year ago

6 0

Step-by-step explanation:

$\frac{(x - 2) {}^{2} }{4} + \frac{(y + 5) {}^{2} }{9} = 1$

This is the equation of the ellipse. Since the denominator is greater for the y values, we have a vertical ellipse. Remember a>b, so a

The formula for the foci of the vertical ellipse is

(h,k+c) and (h,k-c).

where c is

Our center (h,k) is (2, -5)

${c}^{2} = {a}^{2} - {b}^{2}$

Here a^2 is 9, b^2 is 4.

${c}^{2} = 9 - 4$

${c}^{2} = 5$

$c = \sqrt{5}$

So our foci is

$(2, - 5 + \sqrt{5} )$

and

$(2, - 5 - \sqrt{5} )$

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The profile of the cables on a suspension bridge may be modeled by a parabola. The central span of the bridge is 1210 m long and

brilliants [131]

Answer:

The approximated length of the cables that stretch between the tops of the two towers is 1245.25 meters.

Step-by-step explanation:

The equation of the parabola is:

$y=0.00035x^{2}$

Compute the first order derivative of y as follows:

$y=0.00035x^{2}$

$\frac{\text{d}y}{\text{dx}}=\frac{\text{d}}{\text{dx}}[0.00035x^{2}]$

$=2\cdot 0.00035x\\\\=0.0007x$

Now, it is provided that |x | ≤ 605.

⇒ -605 ≤ x ≤ 605

Compute the arc length as follows:

$\text{Arc Length}=\int\limits^{x}_{-x} {1+(\frac{\text{dy}}{\text{dx}})^{2}} \, dx$

$=\int\limits^{605}_{-605} {\sqrt{1+(0.0007x)^{2}}} \, dx \\\\={\displaystyle\int\limits^{605}_{-605}}\sqrt{\dfrac{49x^2}{100000000}+1}\,\mathrm{d}x\\\\={\dfrac{1}{10000}}}{\displaystyle\int\limits^{605}_{-605}}\sqrt{49x^2+100000000}\,\mathrm{d}x\\\\$

Now, let

$x=\dfrac{10000\tan\left(u\right)}{7}\\\\\Rightarrow u=\arctan\left(\dfrac{7x}{10000}\right)\\\\\Rightarrow \mathrm{d}x=\dfrac{10000\sec^2\left(u\right)}{7}\,\mathrm{d}u$

$\int dx={\displaystyle\int\limits}\dfrac{10000\sec^2\left(u\right)\sqrt{100000000\tan^2\left(u\right)+100000000}}{7}\,\mathrm{d}u$

$={\dfrac{100000000}{7}}}{\displaystyle\int}\sec^3\left(u\right)\,\mathrm{d}u\\\\=\dfrac{50000000\ln\left(\tan\left(u\right)+\sec\left(u\right)\right)}{7}+\dfrac{50000000\sec\left(u\right)\tan\left(u\right)}{7}\\\\=\dfrac{50000000\ln\left(\sqrt{\frac{49x^2}{100000000}+1}+\frac{7x}{10000}\right)}{7}+5000x\sqrt{\dfrac{49x^2}{100000000}+1}$

Plug in the solved integrals in Arc Length and solve as follows:

$\text{Arc Length}=\dfrac{5000\ln\left(\sqrt{\frac{49x^2}{100000000}+1}+\frac{7x}{10000}\right)}{7}+\dfrac{x\sqrt{\frac{49x^2}{100000000}+1}}{2}|_{limits^{605}_{-605}}\\\\$

$=1245.253707795227\\\\\approx 1245.25$

Thus, the approximated length of the cables that stretch between the tops of the two towers is 1245.25 meters.

7 0

3 years ago

Find f(x) if it is known that f(x−6)=12x−72.

zheka24 [161]

Answer:

f(x)=12x

Step-by-step explanation

We know that f(x) is our y-coordinate and x is the x-coordinate

Next we should insert x-6 as x in the equation

This is f(x)=12(x-6)-72

But if you look at this that leads to 12x-72 (ignore y-intercept)

That is our first equation

This means that if you take out -6 from the x

You will get f(x)=12x

(Also do you go to RSM?)

5 0

3 years ago

It is one more than the sum of the first three natural numbers, followed by the 24th letter of the alphabet. Please help

Luda [366]

Answer:

The set of natural numbers is the set of all positive integers, then this set is:

{1, 2, 3, ...}

and the 24th letter of the alphabet is x

Now we want to write the expression given in the sentence "It is one more than the sum of the first three natural numbers, followed by the 24th letter of the alphabet"

We can "break" this in parts, so it is easier to understand.

Then:

"...the sum of the first 3 natural numbers..."

is:

1 + 2 + 3

Then:

"...the sum of the first 3 natural numbers, followed by the 24th letter of the alphabet"

This can be written as:

(1 + 2 + 3) + x

Now we can analyze the complete sentence:

"It is one more than the sum of the first three natural numbers, followed by the 24th letter of the alphabet"

This is equal to the expression we found above plus one, then we can write this as:

[(1 + 2 + 3) + x] + 1

[6 + x] + 1

6 + x + 1

7 + x

4 0

3 years ago

Solve for the indicated variable. y = mx + b for x x=

kogti [31]

Answer:

y-b/m = x

Step-by-step explanation:

y = mx + b

~Subtract b to both sides

y - b = mx

~Divide m to both sides

y-b/m = x

Best of Luck!

3 0

3 years ago

How do I find the volume of this need help

Irina-Kira [14]

To find the volume, you need to multiply l x w x h

5 x 7 x 10

5 x 7 = 35

35 x 10 = 350

350 is the correct answer

Glad I could help, and good luck!

AnonymousGiantsFan

5 0

3 years ago