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Sonbull [250]
3 years ago
14

Use the distributive or FOIL method to find the product. (5i + 8)(4i + 6)

Mathematics
1 answer:
balandron [24]3 years ago
3 0
I think you would combine like Terms
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The scale on a map is 1 : 500000 <br> Two towns are 7cm apart on the map
KengaRu [80]

1 : 500,000

multiplied by 7 cm is

7 cm : 3,500,000 cm

= 7 cm : 35000.00 m . . . (100 cm in 1 m)

= 7 cm : 35 km . . . (1000 m in 1 km)

The actual distance between the towns is 35 km.

5 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
Please help me with this question....​
hichkok12 [17]

Answer:

Person D

Step-by-step explanation: Because when you subtract the deductions from what the people earned. Person D has the least amount of money.

7 0
3 years ago
HELP ASAP!! K12 / KEYSTONE QUESTION! I WILL GIVE BRAINLIEST!! HELP!
vladimir1956 [14]

Answer:

A) mean = 1.2

B) median

C) The measures of center use data points to approximate a middle value or average of a given data set

Step-by-step explanation:

The “balance” process was developed to provide another way in which the mean characterizes the “center” of a distribution. 

The mean is the balance point of the data set when the data are shown as dots on a dot plot (or pennies on a ruler). 

A) The balance point for the points 0.4, 1.4, and 1.8. Will be

(0.4 + 1.4 + 1.8)/3 = 3.6/3 = 1.2

B) The median is the measure of center that is indicated by the center of balance

The median is the value in the center of the data. Half of the values are less than the median and half of the values are more than the median. It is probably the best measure of center to use in a skewed distribution.

C) A measure of central (measure of center tendency) is a value that describe a set of data by identifying the central position of the data set. The three measures of central tendency are the mean, median and mode.

5 0
3 years ago
Find the value of n in the solution to the system of equations shown : m=5n-3, m=3n+7 ​
Triss [41]

The value of n is 5

n = 5

4 0
3 years ago
Read 2 more answers
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