Does anyone know the answer to this¿?¿

Among all right triangles whose hypotenuse has a length of 12 cm, what is the largest possible perimeter?

Veronika [31]

Answer:

Largest perimeter of the triangle =

$P(6\sqrt{2}) = 6\sqrt{2} + \sqrt{144-72} + 12 = 12\sqrt{2} + 12 = 12(\sqrt2 + 1)$

Step-by-step explanation:

We are given the following information in the question:

Right triangles whose hypotenuse has a length of 12 cm.

Let x and y be the other two sides of the triangle.

Then, by Pythagoras theorem:

$x^2 + y^2 = (12)^2 = 144\\y^2 = 144-x^2\\y = \sqrt{144-x^2}$

Perimeter of Triangle = Side 1 + Side 2 + Hypotenuse.

$P(x) = x + \sqrt{144-x^2} + 12$

where P(x) is a function of the perimeter of the triangle.

First, we differentiate P(x) with respect to x, to get,

$\frac{d(P(x))}{dx} = \frac{d(x + \sqrt{144-x^2} + 12)}{dx} = 1-\displaystyle\frac{x}{\sqrt{144-x^2}}$

Equating the first derivative to zero, we get,

$\frac{dP(x))}{dx} = 0\\\\1-\displaystyle\frac{x}{\sqrt{144-x^2}} = 0$

Solving, we get,

$1-\displaystyle\frac{x}{\sqrt{144-x^2}} = 0\\\\x = \sqrt{144-x^2}}\\\\x^2 = 144-x^2\\\\x = \sqrt{72} = 6\sqrt{2}$

Again differentiation P(x), with respect to x, using the quotient rule of differentiation.

$\frac{d^2(P(x))}{dx^2} = \displaystyle\frac{-(144-x^2)^{\frac{3}{2}}-x^2}{(144-x)^{\frac{3}{2}}}$

At x = $6\sqrt{2}$ ,

$\frac{d^2(V(x))}{dx^2} < 0$

Then, by double derivative test, the maxima occurs at x = $6\sqrt{2}$

Thus, maxima occurs at x = $6\sqrt{2}$ for P(x).

Thus, largest perimeter of the triangle =

$P(6\sqrt{2}) = 6\sqrt{2} + \sqrt{144-72} + 12 = 12\sqrt{2} + 12 = 12(\sqrt2 + 1)$

7 0

3 years ago

If you are also able to explain this please do! I'd greatly appreciate it!

pogonyaev

You're given a table of x values and an equation. To solve, plug in the values for x.

First box: y=2(-3)-1=-7
Second box: y=2(-2)-1=-5.

See if you can solve the third one.

7 0

3 years ago

Read 2 more answers

Which expression is equivalent to 7h^2-252k^2?

gulaghasi [49]

Answer:

c

Step-by-step explanation:

252 = 7 *36

36 = 6 *6 = 6²

a² -b² = (a + b)(a - b)

7h² - 252k² = 7(h² - 36k²)

= 7(h² - [6k]² )

= 7(h +6k)(h - 6k)

6 0

3 years ago

Read 2 more answers

Tacoma's population in 2000 was about 200 thousand, and has been growing by about 9% each year. If this

frozen [14]

Answer (prediction, first edit):

5,600,000

Equation:

$200,000+(18,000*200)=5,600,000$

7 0

3 years ago

Read 2 more answers

Will give brainlest answer pls help!

mojhsa [17]

You can use 2 step equations to solve real-world problems by assessing the situation, trial and error, As in for example;
if you cannot open a twist off bottle cap by turning it to the right. You have two options twisted to the left to loosen or turn it to the right and not open it”

“ Lefty Lucy righty tightly”

Or if you are married and your husband cheats on you you can do one or two of the following stay or a go

For example stay and be an idiot or go and love yourself and take everything he has

8 0

3 years ago