1) A rancher wants to enclose two rectangular areas near a river, one for sheep and one for cattle. There is 240m of fencing available. Express the area of the enclosures as a function of its dimension.
2) A container company is designing on open-top, square-based, rectangular box that will have a volume of 62.5 cm^3 [cubed]. Express the surface area of the box as a function of its width.
3) An apple farm yields an average of 30 bushels of apples per tree when 20 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases 1 bushel per tree due to the extra congestion. Express the total yield as a function of the number of trees planted.
4) Express the area of an equilateral triangle as a function of the length of one side.
This is for calculus and its a practice assignment. I need a how-to on solving these.
Each answer must be a function.
it would be appreciated
1) To find the area of the enclosures, we need to express the dimensions of the rectangles as variables. Let's assume the dimensions of the sheep enclosure are length L1 and width W1, and the dimensions of the cattle enclosure are length L2 and width W2.
For the sheep enclosure:
Perimeter = 2L1 + 2W1
Fencing used = Perimeter = 2L1 + 2W1
For the cattle enclosure:
Perimeter = 2L2 + 2W2
Fencing used = Perimeter = 2L2 + 2W2
Given that the total fencing available is 240m, we can write the equation:
2L1 + 2W1 + 2L2 + 2W2 = 240
Now, we need to express the area of each enclosure as a function of its dimensions. The area of a rectangle is given by:
Area = Length x Width
For the sheep enclosure:
Area_sheep = L1 x W1
For the cattle enclosure:
Area_cattle = L2 x W2
Thus, the area of the enclosures can be expressed as a function of their dimensions.
2) For the open-top, square-based, rectangular box, let's assume the width of the box is represented by variable W. Since it is square-based, the length and height of the box will also be equal to W.
Given that the volume of the box is 62.5 cm^3, we have:
Volume = Length x Width x Height
62.5 = W x W x W
62.5 = W^3
Now, we need to express the surface area of the box as a function of its width.
The surface area of an open-top box consists of the area of the base and the four sides, excluding the top.
Area_base = Width x Width = W^2
Area_sides = 4 x Length x Height = 4 x W x W = 4W^2
Surface Area = Area_base + Area_sides
Substituting the value of W^3 from the volume equation, we get:
Surface Area = W^2 + 4W^2 = 5W^2
Thus, the surface area of the box can be expressed as a function of its width.
3) Let's assume the number of trees planted per acre is represented by variable N.
Given that the apple farm yields an average of 30 bushels of apples per tree with 20 trees planted on an acre of ground, we can write:
Total Yield = Yield per tree x Number of trees
Total Yield = (30 - (N-20)) x N
Here, 30 represents the initial yield per tree for 20 trees per acre, and (N-20) represents the decrease in yield due to congestion as each additional tree is planted.
Thus, the total yield can be expressed as a function of the number of trees planted.
4) To express the area of an equilateral triangle as a function of the length of one side, let's assume the length of one side is represented by variable S.
The formula for the area of an equilateral triangle is:
Area = (√3 / 4) x (Side length)^2
Substituting the value of S into the formula, we get:
Area = (√3 / 4) x S^2
Thus, the area of the equilateral triangle can be expressed as a function of the length of one side.
looks like "homework dumping" or an assignment is due looking at the different types of questions.
Exactly where are your difficulties?
What have you got so far?