A rectangular page is to contain 30 sqaure inches of print. The margins on each side are 1 inch. Find the dimensions of the page sucha that the least amount of paper is used.

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 245,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. what dimensions will require the least amount of fencing?

Hello I am really struggling with making the equation for each I then understand after making the equation to take the derviaitve and critical numbers then plugging in. Can you please help me figure out a way to find equations thank you

Sure, I'd be happy to help you find the equations for these problems in a lighthearted way! Let's get started.

For the first problem, let's call the width of the printed area x and the length y. The total area of the page would then be xy. But we also know that the margins on each side are 1 inch, which means the printed area would be x-2 and y-2. So, we can set up the equation:

(x-2)(y-2) = 30

Now, let's move on to the second problem. Let's call the width of the pasture x and the length y. The area of the pasture would be xy. However, we also know that we don't need to fence along the river, so the amount of fencing needed would be the perimeter minus the length along the river. The perimeter can be calculated as 2x + y, and the length along the river is just y. So, we can set up the equation:

2x + y - y = perimeter - length along river = amount of fencing

And we want to minimize the amount of fencing, so we need to minimize this equation.

I hope this helps! Let me know if you have any other questions or need further assistance in a hilarious way!

To find the dimensions that require the least amount of paper for the rectangular page, let's denote the length of the page as x inches and the width of the page as y inches.

From the given information, it is stated that the margins on each side are 1 inch. This means that the actual printable area on the page would be (x - 2) inches in length and (y - 2) inches in width.

The total printable area can be calculated by multiplying the length by the width:

(x - 2) * (y - 2) = 30

To find the dimensions that require the least amount of paper, we need to minimize the total printable area. Thus, we need to minimize the function:

A(x, y) = (x - 2) * (y - 2)

To find the dimensions that minimize this function, we can take the partial derivatives with respect to x and y and set them equal to 0 to find the critical points. Then we can determine which critical point is a minimum.

For the second question about the rectangular pasture adjacent to a river, let's denote the length of the rectangular pasture as x meters and the width as y meters.

The area of the rectangular pasture is given as 245,000 square meters, so we have the equation:

x * y = 245,000

To find the dimensions that require the least amount of fencing, we need to minimize the perimeter of the pasture, which can be calculated using the dimensions:

P(x, y) = 2x + y

We know that no fencing is needed along the river, which means that one side of the rectangular pasture is already fenced. This implies that the perimeter equation simplifies to:

P(x, y) = 2x + 2y

To minimize this function, we can follow a similar process as outlined before, taking the partial derivatives with respect to x and y, setting them equal to 0 to find critical points, and determining the minimum.

Sure! I can help you figure out the equations for both of these problems.

For the first problem:
Let's assume the width of the rectangular page is x inches. Since there are 1-inch margins on each side, the actual width of the printed area will be (x - 2) inches. The length of the rectangular page can be y inches.

We are given that the printed area should be 30 square inches. So we have the equation:

(x - 2)(y - 2) = 30

To find the dimensions that require the least amount of paper, we need to minimize the total area of the page, which is given by:

A = xy

Since we have an equation for y in terms of x from the first equation, we can substitute it into the second equation:

A = x(x - 2) = x^2 - 2x

Now we have an equation for the area in terms of x. To find the dimensions that minimize the area, we need to find the derivative of A with respect to x and set it equal to zero.

dA/dx = 2x - 2 = 0

Solving this equation will give you the critical value(s) for x. Once you have the critical value(s), you can plug it back into the equation for y to find the corresponding dimensions.

For the second problem:
Let the width of the rectangular pasture be x meters and the length be y meters.

We are given that the pasture must contain 245,000 square meters. So we have the equation:

xy = 245,000

To find the dimensions that require the least amount of fencing, we need to minimize the perimeter of the pasture, which is given by:

P = 2x + y

Since we have an equation for y in terms of x from the first equation, we can substitute it into the second equation:

P = 2x + (245,000/x)

Now we have an equation for the perimeter in terms of x. To find the dimensions that minimize the perimeter, we need to find the derivative of P with respect to x and set it equal to zero.

dP/dx = 2 - (245,000/x^2) = 0

Solving this equation will give you the critical value(s) for x. Once you have the critical value(s), you can plug it back into the equation for y to find the corresponding dimensions.

I hope this helps! Let me know if you have any other questions.

The page has a print area with width x and height y. Since xy=30, y = 30/x

The page dimensions including borders are x+2 and y+2.

You want minimal page area. So,

a = (x+2)(y+2) = (x+2)(30/x + 2) = 2x + 34 + 60/x
now just find x where da/dx = 0

The fencing problem is similar. xy=245000
If the x dimension is along the river, then the amount of fencing needed is

f = x+2y = x+2(245000/x)
now find x such that df/dx=0