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3 years ago

The perimeter of the hallway is 10 4/10 meters. What is the width, in meters, of the hallway?

1 answer:

4 0

So you have two of those--when you add those together, you get "seven and eight-tenths," or 7.8 The 2 shorter sides are what's left to still measure. If the total perimeter is 10.4, you get the shorter sides by: 10.4 - 7.8 = 2.6, but REMEMBER, the 2.6 is BOTH shorter sides added together, so each ONE of them is 1.3 SO... your hallway/rectangle is 3.9 meters on each long side, and 1.3 meters on each short side. i hope this helps you

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Exactly a year ago, the volume of Arlington lake was 1,620,757 liters. Right now, it’s volume is 382,403 liters greater. How man

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Answer:

Step-by-step explanation: 2003160

If you add them up, you get 2003160.

So the lakes holds 2003160 LITERS of water to the current date.

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2 years ago

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3 years ago

Of the smoothie sold yesterday at Isaac's smoothie shop 1/4 we're banana and another 2/3 were strawberry What fraction of smooth

lesya [120]

Answer:

11/12

Step-by-step explanation:

Fraction of smoothie that is Banana = 1/4

Fraction of smoothie that is strawberry = 2/3

The fraction of smoothies that were either banana or strawberry

= 1/4 + 2/3

Least Common Denominator= 12

= (3 × 1 + 2 × 4)/12

= 3 + 8/12

= 11/12

The fraction of smoothies were either banana or strawberry is 11/12

4 0

3 years ago

EXAMPLE 5 Find the maximum value of the function f(x, y, z) = x + 2y + 11z on the curve of intersection of the plane x − y + z =

Taya2010 [7]

Answer:

$\displaystyle x= -\frac{10}{\sqrt{269}}\\\\\displaystyle y= \frac{13}{\sqrt{269}}\\\\\displaystyle z = \frac{23\sqrt{269}+269}{269}$

<em>Maximum value of f=2.41</em>

Step-by-step explanation:

<u>Lagrange Multipliers</u>

It's a method to optimize (maximize or minimize) functions of more than one variable subject to equality restrictions.

Given a function of three variables f(x,y,z) and a restriction in the form of an equality g(x,y,z)=0, then we are interested in finding the values of x,y,z where both gradients are parallel, i.e.

$\bigtriangledown f=\lambda \bigtriangledown g$

for some scalar $\lambda$ called the Lagrange multiplier.

For more than one restriction, say g(x,y,z)=0 and h(x,y,z)=0, the Lagrange condition is

$\bigtriangledown f=\lambda \bigtriangledown g+\mu \bigtriangledown h$

The gradient of f is

$\bigtriangledown f=$

Considering each variable as independent we have three equations right from the Lagrange condition, plus one for each restriction, to form a 5x5 system of equations in $x,y,z,\lambda,\mu$ .

We have

$f(x, y, z) = x + 2y + 11z\\g(x, y, z) = x - y + z -1=0\\h(x, y, z) = x^2 + y^2 -1= 0$

Let's compute the partial derivatives

$f_x=1\ ,f_y=2\ ,f_z=11\ \\g_x=1\ ,g_y=-1\ ,g_z=1\\h_x=2x\ ,h_y=2y\ ,h_z=0$

The Lagrange condition leads to

$1=\lambda (1)+\mu (2x)\\2=\lambda (-1)+\mu (2y)\\11=\lambda (1)+\mu (0)$

Operating and simplifying

$1=\lambda+2x\mu\\2=-\lambda +2y\mu \\\lambda=11$

Replacing the value of $\lambda$ in the two first equations, we get

$1=11+2x\mu\\2=-11 +2y\mu$

From the first equation

$\displaystyle 2\mu=\frac{-10}{x}$

Replacing into the second

$\displaystyle 13=y\frac{-10}{x}$

Or, equivalently

$13x=-10y$

Squaring

$169x^2=100y^2$

To solve, we use the restriction h

$x^2 + y^2 = 1$

Multiplying by 100

$100x^2 + 100y^2 = 100$

Replacing the above condition

$100x^2 + 169x^2 = 100$

Solving for x

$\displaystyle x=\pm \frac{10}{\sqrt{269}}$

We compute the values of y by solving

$13x=-10y$

$\displaystyle y=-\frac{13x}{10}$

For

$\displaystyle x= \frac{10}{\sqrt{269}}$

$\displaystyle y= -\frac{13}{\sqrt{269}}$

And for

$\displaystyle x= -\frac{10}{\sqrt{269}}$

$\displaystyle y= \frac{13}{\sqrt{269}}$

Finally, we get z using the other restriction

$x - y + z = 1$

Or:

$z = 1-x+y$

The first solution yields to

$\displaystyle z = 1-\frac{10}{\sqrt{269}}-\frac{13}{\sqrt{269}}$

$\displaystyle z = \frac{-23\sqrt{269}+269}{269}$

And the second solution gives us

$\displaystyle z = 1+\frac{10}{\sqrt{269}}+\frac{13}{\sqrt{269}}$

$\displaystyle z = \frac{23\sqrt{269}+269}{269}$

Complete first solution:

$\displaystyle x= \frac{10}{\sqrt{269}}\\\\\displaystyle y= -\frac{13}{\sqrt{269}}\\\\\displaystyle z = \frac{-23\sqrt{269}+269}{269}$

Replacing into f, we get

f(x,y,z)=-0.4

Complete second solution:

$\displaystyle x= -\frac{10}{\sqrt{269}}\\\\\displaystyle y= \frac{13}{\sqrt{269}}\\\\\displaystyle z = \frac{23\sqrt{269}+269}{269}$

Replacing into f, we get

f(x,y,z)=2.4

The second solution maximizes f to 2.4

5 0

3 years ago

The expression 87000(1.12)^t represents the population of a town t hears after 1996. By what percent does the population increas

Tamiku [17]

Answer:

12%

Step-by-step explanation:

Percentage increase of something is often represented in a formula of a format like this:

X=Ar^t

Where X represents the total amount after t years.

A or 87000 in this case represents the starting value, which is this case is the starting population of the town. r or 1.12 represents the percentage increase, if it increases by 12% than the next year has 1.12 times as many people as the previous year so this is represented by 1.12 rather than say 0.12 which would indicate a 88% decrease instead. t is the time, no. of years in this case, that has elapsed since 1996.

This formula basically means that a town population of 87000 increases by 12% for every year of time that elapses.

Hope this helped!

7 0

3 years ago

Read 2 more answers