If you only have a 3-quart pitcher and a 5-quart pitcher, how can you measure exactly 1 quart?

The revenue function is R = -1/20x^2 + 1060x and the profit function is P = -1/20x^2 + 940x - 5000. The value of x that maximizes profit is 9400, and the maximum profit is calculated by substituting this value into the profit function. The price to charge to maximize profit is $590.

a) To express revenue (R) as a function of x, we multiply the price (p) by the quantity (x):

R = p * x

Substituting the demand equation (p = -1/20x + 1060):

R = (-1/20x + 1060) * x

Simplifying:

R = -1/20x^2 + 1060x

b) Profit (P) is given by subtracting the cost (c(x)) from the revenue (R):

P = R - c(x)

Substituting the expressions for R and c(x):

P = (-1/20x^2 + 1060x) - (120x + 5000)

Simplifying:

P = -1/20x^2 + 940x - 5000

c) To find the value of x that maximizes profit, we take the derivative of the profit function with respect to x and set it equal to zero:

dP/dx = -1/10x + 940

Solving for x:

-1/10x + 940 = 0

x = 9400

d) To find the price that maximizes profit, we substitute the value of x (9400) into the demand equation:

p = -1/20(9400) + 1060

simplifying:

p = 1060 - 470

p = $590

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a) First, we must express the revenue, denoted as 'R'. Revenue is simply the price per item multiplied by the quantity of items sold. Given the demand equation 'p', we can express the revenue as follows:

R = p * x = ( -1/20 * x + 1060 ) * x

This simplifies to:

R = x * (1060 - 0.05*x)

b) Profit is revenue minus cost. We use the revenue equation from part a) and the given cost equation 'c = 120x + 5000'. Plug these into the profit equation:

P = R - c = ( x * (1060 - 0.05*x) ) - ( 120x + 5000 )

This simplifies to

P = x * (1060 - 0.05*x) - 120*x - 5000

c) To maximize profit, we need to find the derivative of the profit function and then solve for x when this derivative is equal to zero (this will find the quantity that maximizes profit). After doing this, we find that the quantity that maximizes profit is x = 9400.

If we substitute x = 9400 into the profit function, we get the maximum profit which turns out to be 4413000.

d) The price that should be charged to maximize profit is found by substituting the value of x that gave us the maximum profit into the price function. In this case, substituting x = 9400 gives us the price = 590.

Therefore, in order to maximize profit, 9400 lawnmowers should be produced and sold at a price of 590 per lawnmower, which will yield a profit of 4413000.

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The task is to determine the coordinates of two 2-cm holes in a quarter-circle steel plate, but there is insufficient information to provide a precise location without additional details or constraints.

The question requires us to find the coordinates for the centers of two 2-cm holes to be drilled in a steel plate, which is shaped like a quarter of a circle with a radius of 49 cm. Assuming the plate is positioned with its corner at the origin (0,0), with the circular edge encompassed by the positive x-axis and positive y-axis, the holes should be placed in such a way that their centers within the quarter circle are equidistant from the axes and from the plate's circular edge.

However, the question does not provide sufficient information or constraints about the location of the holes within the quarter-circle plate. To determine the exact coordinates, we need more specifics, such as how far from the edges or any other centering constraints for the holes. Without this additional information, there are infinitely many possible coordinates for the holes that meet the given condition of being within the quarter-circle plate.

If the holes are to be evenly distributed and placed symmetrically, a possible choice could be to align the centers of the holes along the line y = x, which bisects the quarter-circle. Yet, without precise instructions, we cannot provide a definitive answer to the coordinates of the holes.

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Answer:

LM= 4

Step-by-step explanation:

In the triangle KLM, ON is the mid segment of KLM

WE use triangle Mid segment theorem

MN= twice of ON

MN= 2 times of ON

4x= 2(x+1)

distribute 2 inside the parenthesis

4x= 2x+2

Subtract 2x on both sides

4x-2x= 2x+2-2x

2x= 2

divide by 2 on both sides

x= 1

To find length of MN, we plug in 1 for x

MN= 4x= 4(1)= 4

Answer:  D. [tex]\pi=\dfrac{C}{2r}[/tex]

Step-by-step explanation:

Given : The formula to find the circumference is given by :-

[tex]C=2\pi r[/tex], where r is the radius of the circle.

To find the formula for [tex]\pi[/tex] , we need to divide 2r on both of the sides in the above equation, we get

[tex]\pi=\dfrac{C}{2r}[/tex] , where  r is the radius of the circle.

Hence, the correct formula for [tex]\pi[/tex] : Option D [tex]\pi=\dfrac{C}{2r}[/tex]

Final answer:

The greatest common factor of the monomials 16x and 36x is 4x, obtained by finding the common prime factors of 16 and 36 and including the variable x, which is present in both monomials.

Explanation:

To find the greatest common factor of the monomials 16x and 36x, we need to break each monomial into its prime factors and compare them.

Prime factors of 16 are 2 x 2 x 2 x 2.

Prime factors of 36 are 2 x 2 x 3 x 3.

Both 16 and 36 have two 2's in common.

Therefore, the GCF of 16 and 36 is 2 x 2 = 4. Since x is present in both monomials, x is also part of their GCF.

The greatest common factor of the monomials 16x and 36x is therefore 4x.

Answer:

the answer to this question is 4x - 3(12x + 4) = -6 hope this helps

The fifth root of -32 is -2. To find it, think of a number that, when raised to the power of 5, gives the original number.

The fifth root of -32 is -2. To find the fifth root of a number, we can think of it as finding a number that, when raised to the power of 5, gives us the original number. In this case, -2 * -2 * -2 * -2 * -2 = -32.

Answer:

[tex]h(x)=\frac{x}{2}-\frac{1}{2}[/tex]

Step-by-step explanation:

Hello

Step1

Let f(x)=y

f(x) = 2x + 1

(1)y= 2x + 1  

to find the inverse of f(x)

Replace every x with a y and replace every y with an x

y= 2x + 1

replacing  

x=2y+1

Step 2

Now solve the equation for y

x=2y+1

subtract 1 on both sides

x-1=2y+1-1

x-1=2y

divide boths dies by 2

(x-1)/2=2y/y

[tex]\frac{x-1}{2} =\frac{2y}{2} \\y=\frac{x-1}{2}[/tex]

let this last equation h(x)= inverse of f(x)

[tex]h(x)=\frac{x-1}{2}\\h(x)=\frac{x}{2}-\frac{1}{2}[/tex]

Have a great day

2(b−4)=6(b) 2b - 8 = 6b 2b - 6b = 8 -4b = 8 Now you can solve.....

Answer:

1st one : A′(-3, 3), B′(-1, 1), C′(-2, 3)

Step-by-step explanation:

below

The surface area of an open-ended pipe with a radius of 6 inches and a height of 18 inches is 678.24 square inches. The calculation is based on the formula for the circumference of the circle, which is multiplied by the height of the cylinder.

To calculate the surface area of a pipe that is open at both ends, we need to find the area of the outer surface. Since the pipe is a cylinder without the top and bottom circles, the surface area is simply the circumference of the circle (which is the edge of the openings) times the height of the cylinder. The formula for the circumference of a circle is C = 2πr, where r is the radius.

The radius is given as 6 inches and the height is 18 inches, therefore:

The cone of dimension 4.5 inches height and 1.5 inches radius can hold 9.425 inches^3 of ice cream.

A sugar cone can be seen as a three-dimensional object, specifically a cone, which has a known formula for calculating its volume.

Volume of a cone: (1/3) x π x radius2 x height

Substituting the given values: (1/3) x π x 1.52 x 4.5 = 9.425 inches^3.