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3 years ago

A(4,-7) B(-2,1) find AB

1 answer:

8 0

$\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\A(\stackrel{x_1}{4}~,~\stackrel{y_1}{-7})\qquadB(\stackrel{x_2}{-2}~,~\stackrel{y_2}{1})\qquad \qquadd = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}\\\\\\AB=\sqrt{[-2-4]^2+[1-(-7)]^2}\implies AB=\sqrt{(-2-4)^2+(1+7)^2}\\\\\\AB=\sqrt{(-6)^2+8^2}\implies AB=\sqrt{36+64}\implies AB=\sqrt{100}\implies AB=10$

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The number 42 is increased to 46. what is the percentage by which the number was increased?

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Answer:

9.52% ( to 2 dec. places )

Step-by-step explanation:

Calculate the percentage increase using

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increase = 46 - 42 = 4, hence

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2 years ago

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Bc has endpoints b(5 ,9) and c(-4, -3). Find the coordinates of the midpoint of bc

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3 years ago

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Three vertices of a rectangle are located on the coordinate plane shown below. Coordinate plane with points labeled Part A Which

Harlamova29_29 [7]

Answer:

Part A

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A. (2, 8)

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D. (8, 6)

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Which point is at (8, 2)?

A. A

B. B

C. C

D. None of these

Part C

What are the coordinates of point A?

Enter your answer in the boxes.

(

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Part D

What are the coordinates of the fourth vertex of the rectangle?

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3 years ago

Use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinat

tensa zangetsu [6.8K]

Answer:

The volume of the largest rectangular box (V) = 81/4

Step-by-step explanation:

Step 1:-

Given volume of the largest rectangular box in the first octant

V = l b h

let (x ,y, z) be the one vertex in the given plane

V = f(x, y, z) = x y z

given plane. Ф (x, y, z) = x + 9y + 4z = 27 ........(1)

Step( ii):

By using Lagrange multipliers

Suppose it is required to find the extreme for the function f(x, y, z) subject to the condition Ф (x, y, z) =0

Form Lagrange function F(x, y, z) = f(x, y, z) + λ Ф (x, y, z) where λ is called the Lagrange multipliers

F(x, y, z) = x y z+ λ ( x + 9y + 4z - 27) ......(2)

Obtain the equations are δ F / δ x = 0

δ f / δ x +λ δ Ф / δ x =0

⇒ y z + λ ( 1) = 0

⇒ y z = - λ ......(a)

Obtain the equations are δ F / δ y = 0

δ f / δ x +λ δ Ф / δ x =0

⇒ x z + λ ( 9) = 0

⇒ $\frac{xz}{9}$ = - λ .......(b)

Obtain the equations are δ F / δ z = 0

δ f / δ x +λ δ Ф / δ z =0

x y + λ ( 4) = 0

⇒ $\frac{xy}{4}$ = - λ .......(c)

Step (iii):-

Equating (a) and (b) equations

we get $y z = \frac{xz}{9}$

cancel 'z' value we get x = 9y .......(d)

Equating (b) and (c) equations

we get $\frac{xy}{4} = \frac{xz}{9}$

cancel 'x' value on both sides , we get

4z =9y ......(e)

substitute (d) (e) values in equation (1) we get

x + 9y + 4z -27 = 0

9y + 9y + 9y -27 =0

27y -27 =0

y = 1

substitute y =1 in x = 9y

x = 9

substitute y =1 in 4z =9y

z = 9 /4

therefore the dimensions are x =9 , y=1 and z = 9 /4

Conclusion:-

The largest volume of the rectangular box V = x y z

substitute x =9 , y=1 and z = 9 /4

V = 9(1)(9/4) =81/4

The largest volume of the rectangular box (V) = 81/4

Verification :-

Given plane x + 9y + 4z = 27

substitute x =9 , y=1 and z = 9 /4

9 + 9 + 4(9/4) = 27

27 =27

so satisfied equation

3 0

3 years ago