Which fraction represents the product of -7/8 and -9/11 in simplest form?

Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ell

Tanzania [10]

Answer:

Length (parallel to the x-axis): $2 \sqrt{2}$ ;

Height (parallel to the y-axis): $4\sqrt{2}$ .

Step-by-step explanation:

Let the top-right vertice of this rectangle $(x,y)$ . $x, y >0$ . The opposite vertice will be at $(-x, -y)$ . The length the rectangle will be $2x$ while its height will be $2y$ .

Function that needs to be maximized: $f(x, y) = (2x)(2y) = 4xy$ .

The rectangle is inscribed in the ellipse. As a result, all its vertices shall be on the ellipse. In other words, they should satisfy the equation for the ellipse. Hence that equation will be the equation for the constraint on x and y.

For Lagrange's Multipliers to work, the constraint shall be in the form: $g(x, y) =k$ . In this case

$\displaystyle g(x, y) = \frac{x^{2}}{4} + \frac{y^{2}}{16}$ .

Start by finding the first derivatives of $f(x, y)$ and $g(x, y)$ with respect to x and y, respectively:

$f_x = y$ ,
$f_y = x$ .

$\displaystyle g_x = \frac{x}{2}$ ,
$\displaystyle g_y = \frac{y}{8}$ .

This method asks for a non-zero constant, $\lambda$ , to satisfy the equations:

$f_x = \lambda g_x$ , and

$f_y = \lambda g_y$ .

(Note that this method still applies even if there are more than two variables.)

That's two equations for three variables. Don't panic. The constraint itself acts as the third equation of this system:

$g(x, y) = k$ .

$\displaystyle \left\{ \begin{aligned} &y = \frac{\lambda x}{2} && (a)\\ &x = \frac{\lambda y}{8} && (b)\\ & \frac{x^{2}}{4} + \frac{y^{2}}{16} = 1 && (c)\end{aligned}\right.$ .

Replace the $y$ in equation $(b)$ with the right-hand side of equation $(b)$ .

$\displaystyle x = \lambda \frac{\lambda \cdot \dfrac{x}{2}}{8} = \frac{\lambda^{2} x}{16}$ .

Before dividing both sides by $x$ , make sure whether $x = 0$ .

If $x = 0$ , the area of the rectangle will equal to zero. That's likely not a solution.

If $x \neq 0$ , divide both sides by $x$ , $\lambda = \pm 4$ . Hence by equation $(b)$ , $y = 2x$ . Replace the $y$ in equation $(c)$ with this expression to obtain (given that $x, y >0$ ) $x = \sqrt{2}$ . Hence $y = 2x = 2\sqrt{2}$ . The length of the rectangle will be $2x = 2\sqrt{2}$ while the height will be $2y = 4\sqrt{2}$ . If there's more than one possible solutions, evaluate the function that needs to be maximized at each point. Choose the point that gives the maximum value.

7 0

3 years ago

Find a formula for the inverse of the function f(x)=3x^3 + 2sinx + 4cosx

vitfil [10]

Explicit Functiony = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation and does not appear at all on the other side. (ex. y = -3x + 5)Implicit FunctionAn equation in which y is not alone on one side. (ex. 3x + y = 5)Implicit DifferentiationGiven a relation of x and y, find dy/dx algebraically.d/dx ln(x)1/xd/dx logb(x) (base b)1/xln(b)d/dx ln(u)1/u × du/dxd/dx logb(u) (base b)1/uln(b) × du/dx(f⁻¹)'(x) = 1/(f'(f⁻¹(x))) iff is a differentiable and one-to-one functiondy/dx = 1/(dx/dy) ify = is a differentiable and one-to-one functiond/dx (b∧x)b∧x × ln(b)d/dx e∧xe∧xd/dx (b∧u)b∧u × ln(b) du/dxd/dx (e∧u)e∧u du/dxDerivatives of inverse trig functionsStrategy for Solving Related Rates Problems1. Assign letters to all quantities that vary with time and any others that seem relevant to the problem. Give a definition for each letter.

2. Identify the rates of change that are known and the rate of change that is to be found. Interpret each rate as a derivative.

3. Find an equation that relates the variables whose rates of change were identified in Step 2. To do this, it will often be helpful to draw an appropriately labeled figure that illustrates the relationship.

4. Differentiate both sides of the equation obtained in Step 3 with respect to time to produce a relationship between the known rates of change and the unknown rate of change.

5. After completing Step 4, substitute all known values for the rates of change and the variables, and then solve for the unknown rate of change.Local Linear Approximation formulaf(x) ≈ f(x₀) + f'(x₀)(x - x₀)
f(x₀ + ∆x) ≈ f(x₀) + f'(x₀)∆x when ∆x = x - x₀Local Linear Approximation from the Differential Point of View∆y ≈ f'(x)dx = dyError Propagation Variablesx₀ is the exact value of the quantity being measured
y₀ = f(x₀) is the exact value of the quantity being computed
x is the measured value of x₀
y = f(x) is the computed value of yL'Hopital's RuleApplying L'Hopital's Rule1. Check that the limit of f(x)/g(x) is an indeterminate form of type 0/0.
2. Differentiate f and g separately.
3. Find the limit of f'(x)/g'(x). If the limit is finite, +∞, or -∞, then it is equal to the limit of f(x)/g(x).

5 0

4 years ago

What is the solution to the equation P = 2L + 2w when solved for w?

ira [324]

Answer: w=(P - 2L)/2

Step-by-step explanation:

P=2L+2W

P - 2L=2W

Divide both sides by 2

(P - 2L)/2=2W/2

(P - 2L)/2=W

W=(P - 2L)/2

8 0

3 years ago

The distance from 2nd base to home plate.

Ratling [72]

Answer:

127.3 ft

Step-by-step explanation:

Make it a right triangle...

ΔABC is 90 by 90 by x

x = $\sqrt{90^2+90^2}$

x = $\sqrt{8100+8100}$

x = $\sqrt{16200}$

x = 127.3 ft

4 0

3 years ago

BRAINLIESSTTTT ASAP !!!!!!!!!! 20 pointssss

Mars2501 [29]

Answers:
_____________________________________________________
Part A) " (3x + 4) " units .
_____________________________________________________
Part B) "The dimensions of the rectangle are:

" (4x + 5y) " units ; AND: " (4x − 5y)" units."
_____________________________________________________

Explanation for Part A):
_____________________________________________________

Since each side length of a square is the same;

Area = Length * width = L * w ; L = w = s = s ;

in which: " s = side length" ;

So, the Area of a square, "A" = L * w = s * s = s² ;

{Note: A "square" is a rectangle with 4 (four) equal sides.}.

→ Each side length, "s", of a square is equal.

Given: s² = "(9x² + 24x + 16)" square units ;

Find "s" by factoring: "(9x² + 24x + 16)" completely:

→ " 9x² + 24x + 16 ";

Factor by "breaking into groups" :

"(9x² + 24x + 16)" =

 → "(9x² + 12x) (12x + 16)" ;
_______________________________________________________

Given: " (9x² + 24x + 16) " ;
_______________________________________________________
Let us start with the term:
_______________________________________________________

" (9x² + 12x) " ;

 → Factor out a "3x" ; → as follows:
_______________________________________

 → " 3x (3x + 4) " ;

Then, take the term:
_______________________________________
→ " (12x + 16) " ;

And factor out a "4" ; → as follows:
_______________________________________

→ " 4 (3x + 4) "
_______________________________________
We have:

" 9x² + 24x + 16 " ;

= " 3x (3x + 4) + 4(3x + 4) " ;
_______________________________________
Now, notice the term: "(3x + 4)" ;

We can "factor out" this term:

3x (3x + 4) + 4(3x + 4) =

→ " (3x + 4) (3x + 4) " . → which is the fully factored form of:

" 9x² + 24x + 16 " ;
____________________________________________________
→ Or; write: " (3x + 4) (3x + 4)" ; as: " (3x + 4)² " .
____________________________________________________
→ So, "s² = 9x² + 24x + 16 " ;

Rewrite as: " s² = (3x + 4)² " .

→ Solve for the "positive value of "s" ;

→ {since the "side length of a square" cannot be a "negative" value.}.
____________________________________________________
→ Take the "positive square root of EACH SIDE of the equation;
to isolate "s" on one side of the equation; & to solve for "s" ;

→ ⁺√(s²) = ⁺√[(3x + 4)²] '

To get:

→ s = " (3x + 4)" units .
_______________________________________________________

Part A): The answer is: "(3x + 4)" units.
____________________________________________________

Explanation for Part B):
_________________________________________________________

The area, "A" of a rectangle is:

A = L * w ;

in which "A" is the "area" of the rectangle;
"L" is the "length" of the rectangle;
"w" is the "width" of the rectangle;
_______________________________________________________
Given: " A = (16x² − 25y²) square units" ;

→ We are asked to find the dimensions, "L" & "w" ;
→ by factoring the given "area" expression completely:
____________________________________________________
→ Factor: " (16x² − 25y²) square units " completely '

Note that: "16" and: "25" are both "perfect squares" ;

We can rewrite: " (16x² − 25y²) " ; as:

= " (4²x²) − (5²y²) " ; and further rewrite the expression:
________________________________________________________
Note:
________________________________________________________
" (16x²) " ; can be written as: "(4x)² " ;

↔ " (4x)² = "(4²)(x²)" = 16x² "

Note: The following property of exponents:

→ (xy)ⁿ = xⁿ yⁿ ; → As such: " 16x² = (4²x²) = (4x)² " .
_______________________________________________________
Note:
_______________________________________________________

→ " (25x²) " ; can be written as: " (5x)² " ;

↔ "( 5x)² = "(5²)(x²)" = 25x² " ;

Note: The following property of exponents:

→ (xy)ⁿ = xⁿ yⁿ ; → As such: " 25x² = (5²x²) = (5x)² " .
______________________________________________________

→ So, we can rewrite: " (16x² − 25y²) " ;

as: " (4x)² − (5y)² " ;

→ {Note: We substitute: "(4x)² " for "(16x²)" ; & "(5y)² " for "(25y²)" .} . ;
_______________________________________________________
→ We have: " (4x)² − (5y)² " ;

→ Note that we are asked to "factor completely" ;

→ Note that: " x² − y² = (x + y) (x − y) " ;

→ {This property is known as the "difference of squares".}.

→ As such: " (4x)² − (5y)² " = " (4x + 5y) (4x − 5y) " .
_______________________________________________________
Part B): The answer is: "The dimensions of the rectangle are:

" (4x + 5y) " units ; AND: " (4x − 5y)" units."
_______________________________________________________

7 0

3 years ago