Question 1

Evaluate $f ( a , - 3 )$ for
​ $f ( x , y ) = x ^ { 2 } - y - x y + 6$

Accepted Answer

The answer of Evaluate \(f ( a , - 3...

Question 2

Cobb-Douglas productivity functions have the form
  $p ( x , y ) = k x ^ { a } y ^ { 1 - a }$  
where P stands for the number of items produced per year, x is the number of employees, and y is the annual operating budget. The numbers k and a are constants that depend on the situation we are looking at, with $0 \leq a \leq 1$ .
 
How many items will be produced per year by a company that has 250 employees and an annual operating budget of $100,000 if $k = 1,000$ and $a = 0.5$   (Round your answer to one significant digit.)

Accepted Answer

The answer of Cobb-Douglas productivity functions have the form
 ...

Question 3

Your on-line bookstore is in direct competition with Amazon.com, BN.com, and Borders.com. Your company's daily revenue in dollars is given by   $R ( x , y , z ) = 25,000 - 0.02 x - 0.02 y - 0.01 z + 0.00005 y z$  
Where x, y, and z are the online daily revenues of Amazon.com, BN.com, and Borders.com, respectively. If Amazon.com and BN.com each show a daily revenue of $1,000, give an equation showing how your daily revenue depends on that of Borders.com.
 
A)R (z) = 24,960 + 0.04z
B) R (z) = 25,000 - 20x - 20y - 0.01z + 0.00005yz
C) R (z) = 25,000 - 0.01z
D) R (z) = 24,960
E) R (z) = 25,000 - 0.01z + 0.00005z

Accepted Answer

When x and y are both 1000, the revenue function becomes:R(z) = 25,000 - 0.02(1000) - 0.02(1000) - 0.01z + 0.00005(1000)(z)= 25,000 - 20 - 20 - 0.01z + 0.5z= 24,960 + 0.04z

Question 4

Use the given tabular representation of the function f to compute f (5, 10) - f (10, 5).
​ $$\begin{array} { | l | l | l | l | l | l | } 
\hline & & \boldsymbol { x } & & & \
\hline & & \mathbf { 5 } & \mathbf { 1 0 } & \mathbf { 1 5 } & \mathbf { 2 0 } \
\hline \boldsymbol { y } & \mathbf { 5 } & - 1 & 105 & 151 & - 5 \
\hline & \mathbf { 1 0 } & - 1 & 179 & 117 & - 10 \
\hline & \mathbf { 1 5 } & - 11 & 216 & 255 & - 16 \
\hline & \mathbf { 2 0 } & - 12 & 257 & 211 & - 15 \
\hline
\end{array}$$

Accepted Answer

The answer of Use the given tabular representation of the...

Question 5

Let $f ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 z$ .
 
By what value is $f$ increased if $y$ is increased by 1

Accepted Answer

The answer of Let \(f ( x , y ,...

Question 6

The burden of human made aerosol sulfate in the earth's atmosphere, in grams per square meter, is   $B ( x , n ) = \frac { x n } { A }$  
Where x is the total weight of aerosol sulfate emitted into the atmosphere per year and n is the number of years it remains in the atmosphere. A is the surface area of the earth, approximately $5 \times 10 ^ { 14 }$ square meters. Calculate the burden given the 1995 estimated values of $x = 2.8 \times 10 ^ { 14 }$ grams per year and $n = 20$ days. Enter the number of grams per square meter as a number without the units. Round to three decimal places.
 
A)0.061 grams per square meter
B) 0.098 grams per square meter
C) 0.031 grams per square meter
D) 0.767 grams per square meter
E) 11.200 grams per square meter

Accepted Answer

To calculate the burden of human-made aerosol sulfate in the earth's atmosphere, we use the given formula $B ( x , n ) = \frac { x n } { A }$. Given $x = 2.8 \times 10 ^ { 14 }$ grams per year, $n = 20$ days (which needs to be converted to years for consistency, so $n = \frac{20}{365} \approx 0.0548$ years), and $A = 5 \times 10 ^ { 14 }$ square meters, we substitute these values into the formula:$$B ( x , n ) = \frac { 2.8 \times 10 ^ { 14 } \times 0.0548 } { 5 \times 10 ^ { 14 }}$$$$B ( x , n ) = \frac { 15.344 \times 10 ^ { 12 } } { 5 \times 10 ^ { 14 }}$$$$B ( x , n ) = 0.030688 $$Rounded to three decimal places, this gives a burden of 0.031 grams per square meter.

Question 7

Graph the function.   $f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }$  
A)   
 
B) 
C)   
 
D)

Accepted Answer

The answer of Graph the function.   \(f (...

Question 8

Cobb-Douglas productivity functions have the form   $p ( x , y ) = k x ^ { a } y ^ { 1 - a }$  
Where P stands for the number of items produced per year, x is the number of employees, and y is the annual operating budget. The numbers k and a are constants that depend on the situation we are looking at, with $0 \leq a \leq 1$ .
 
How many items will be produced per year by a company that has 150 employees and an annual operating budget of $80,000 if $k = 500$ and $a = 0.4$   (Round your answer to one significant digit.)
 
A)3245342.2
B) 4,000,000
C) 4245342.2
D) 1,500,000
E) 3,000,000

Accepted Answer

The answer of Cobb-Douglas productivity functions have the form ...

Question 9

Your weekly cost (in dollars) to manufacture x cars and y trucks is
  $$C ( x , y ) = 270,000 + 6,000 x + 5,000 y$$  
What is the marginal cost of a car 
 
Please enter your answer as a number without the units.

Accepted Answer

The answer of Your weekly cost (in dollars) to manufacture...

Question 10

Choose the correct letter for each question.
-neither
A)$P = 2 + 2 x - z$
B)$L = 4 x - 5 y + 6 x z$
C)$L = 3 x + 6 y + 6 x y - 6 y ^ { 2 }$

Accepted Answer

The answer of Choose the correct letter for each question.
-neither
A)\(P...

Question 11

Evaluate $g ( - 2,1 , - 1 )$ for
​ $g ( x , y , z ) = \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Accepted Answer

The answer of Evaluate \(g ( - 2,1 , -...

Question 12

Choose the correct letter for each question.
-linear
A)$P = 2 + 2 x - z$
B)$L = 4 x - 5 y + 6 x z$
C)$L = 3 x + 6 y + 6 x y - 6 y ^ { 2 }$

Accepted Answer

A linear equation is an equation where the highest power of the variable(s) is 1. Option A, $P = 2 + 2x - z$, is the only equation that meets this criterion, as it does not contain any variables raised to a power higher than 1 or any products of variables.

Question 13

Your weekly cost (in dollars) to manufacture x cars and y trucks is   $C ( x , y ) = 240,000 + 6,000 x + 4,000 y$  
What is the marginal cost of a truck 
 
A)$6,000
B) $4,000
C) $2,000
D) $244,000
E) $246,000

Accepted Answer

The marginal cost of a truck is the additional cost incurred by producing one more truck, which is represented by the coefficient of $y$ in the cost function $C(x, y)$. Therefore, the marginal cost of a truck is $4,000.

Question 14

Suppose your newspaper is trying to decide between two competing desktop publishing software packages, Macro Publish and Turbo Publish. You estimate that if you purchase x copies of Macro Publish and y copies of Turbo Publish, your company's daily productivity will be   $U ( x , y ) = 8 x ^ { 0.9 } y ^ { 0.1 } + x$  
Where U (x, y) is measured in pages per day (U is called a utility function.) If $x = y = 9$ , by approximately how many pages per day will productivity increase if x is increased by one unit 
 
A)A productivity increase of approximately 7.16 pages per day.
B) A productivity decrease of approximately 81.00 pages per day.
C) A productivity increase of approximately 89.16 pages per day.
D) A productivity increase of approximately 16.32 pages per day.
E) A productivity increase of approximately 8.16 pages per day.

Accepted Answer

To find the increase in productivity if $x$ is increased by one unit from $x = y = 9$, we first calculate the productivity at $x = 9$ and $y = 9$, and then calculate it at $x = 10$ and $y = 9$, and find the difference.Original productivity at $x = 9, y = 9$:$$U(9, 9) = 8(9)^{0.9}(9)^{0.1} + 9$$New productivity at $x = 10, y = 9$:$$U(10, 9) = 8(10)^{0.9}(9)^{0.1} + 10$$The increase in productivity is the difference between these two values.Calculating the original productivity:$$U(9, 9) = 8(9)^{0.9}(9)^{0.1} + 9 \approx 8(9)^1 + 9 = 72 + 9 = 81$$Calculating the new productivity:$$U(10, 9) = 8(10)^{0.9}(9)^{0.1} + 10 \approx 8(10) + 10 = 80 + 10 = 90$$The increase in productivity:$$90 - 81 = 9$$However, the calculation above simplifies the process incorrectly by assuming $9^{0.9} \times 9^{0.1} = 9^1$ and $10^{0.9} \times 9^{0.1} = 10$, which is not accurate. The correct approach involves accurately calculating the utility values with the given exponents and then finding the difference. Let's correct the calculation:Correct calculation for original productivity:$$U(9, 9) = 8 \cdot 9^{0.9} \cdot 9^{0.1} + 9$$Correct calculation for new productivity:$$U(10, 9) = 8 \cdot 10^{0.9} \cdot 9^{0.1} + 10$$Given the mistake in the initial explanation, the correct answer involves accurately calculating these expressions and finding the difference, which should approximate the increase in productivity as given in one of the options. The correct approach involves using the utility function accurately with the given values of $x$ and $y$, and the increase in productivity should be calculated based on the change in $x$ from 9 to 10, which would lead to the correct answer choice E, acknowledging an error in the initial calculation process.

Question 15

Evaluate $g ( 0,1,0 )$ for
​ $g ( x , y , z ) = \ln ( x + y + z )$

Accepted Answer

The answer of Evaluate $g ( 0,1,0 )$ for
​ \(g...

Question 16

Your on-line bookstore is in direct competition with Amazon.com, BN.com, and Borders.com. Your company's daily revenue in dollars is given by
​ $R ( x , y , z ) = 15,000 - 0.02 x - 0.01 y - 0.02 z + 0.0001 y z$ ​
where x, y, and z are the online daily revenues of Amazon.com, BN.com, and Borders.com, respectively. If Amazon.com and BN.com each show a daily revenue of $5,000, give an equation showing how your daily revenue depends on that of Borders.com.

Accepted Answer

The answer of Your on-line bookstore is in direct competition...

Question 17

Choose the correct letter for each question.
-interaction
A)$P = 2 + 2 x - z$
B)$L = 4 x - 5 y + 6 x z$
C)$L = 3 x + 6 y + 6 x y - 6 y ^ { 2 }$

Accepted Answer

The answer of Choose the correct letter for each question.
-interaction
A)\(P...

Quiz 15: Functions of Several Variables

Evaluate $f ( a , - 3 )$ for $f ( x , y ) = x ^ { 2 } - y - x y + 6$

Let $f ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 z$ . By what value is $f$ increased if $y$ is increased by 1

Graph the function. $f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Your weekly cost (in dollars) to manufacture x cars and y trucks is $C ( x , y ) = 270,000 + 6,000 x + 5,000 y$ What is the marginal cost of a car Please enter your answer as a number without the units.

Choose the correct letter for each question. -neither

Evaluate $g ( - 2,1 , - 1 )$ for $g ( x , y , z ) = \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Choose the correct letter for each question. -linear

Your weekly cost (in dollars) to manufacture x cars and y trucks is $C ( x , y ) = 240,000 + 6,000 x + 4,000 y$ What is the marginal cost of a truck

Evaluate $g ( 0,1,0 )$ for $g ( x , y , z ) = \ln ( x + y + z )$

Choose the correct letter for each question. -interaction

Quiz 15: Functions of Several Variables

Evaluate f(a,−3)f ( a , - 3 )f(a,−3) for ​ f(x,y)=x2−y−xy+6f ( x , y ) = x ^ { 2 } - y - x y + 6f(x,y)=x2−y−xy+6

Let f(x,y,z)=27−1.4x+6.2y+10zf ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 zf(x,y,z)=27−1.4x+6.2y+10z . By what value is fff increased if yyy is increased by 1

Graph the function. f(x,y) =1.5e−(x2+y2) f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }f(x,y) =1.5e−(x2+y2)

Your weekly cost (in dollars) to manufacture x cars and y trucks is C(x,y)=270,000+6,000x+5,000yC ( x , y ) = 270,000 + 6,000 x + 5,000 yC(x,y)=270,000+6,000x+5,000y What is the marginal cost of a car Please enter your answer as a number without the units.