Suppose that 201 ft of fencing are used to enclose a corral...

Suppose that 201 ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle as the following figure: x y Find the dimensions of the corral with maximum area. x = ft. y = ft. Find the point P on the graph of the function y = ?x closest to the point (9, 0) The x coordinate of P is: . A box is contructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $3 per square foot and the metal for the sides costs $8 per square foot. Find the dimensions that minimize cost if the box has a volume of 50 cubic feet. Length of base x = Height of side z = A boat on the the ocean is 9 km from the nearest point on a straight shoreline; that point is 10 km from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. If she walks at 5 km/h and rows at 5 km/h. How far will the point on shore be from the restaurant if she plans to minimize her total travel time? Distance from restaurant = km Find a positive number x such that the sum of 36x and 1/x is as small as possible. x = Does this problem require optimization over an open interval or a closed interval? A. closed B. open

Transcript

00:01 We have to carry out a number of optimization problems.

00:04 In the first problem we are given a coral whose fencing is shown by this blue line.

00:12 Now this carved part is actually half of a circle.

00:18 Now the area of a circle is given by pi radius r square and its perimeter is given by 2 pi r r.

00:29 So for half of a circle, the area is pi r square by 2 and perimeter is 2 pi r by 2.

00:36 Now here the radius r is given by x by 2 because x is the diameter.

00:43 So the total length of the perimeter is given by p which is equal to 2 pi into radius by 2 plus this vertical parts which is y plus y equal to 2 y and then this horizontal part which is x.

00:59 Now the total perimeter is equal to 201 feet.

01:03 This allows us to solve y in terms of x which takes this form.

01:08 Now let us find out the total area of the coral which is given by a.

01:13 Now a is equal to pi r square by two plus xy because pi r square by two is half the area of the circle and xy is the square area.

01:25 And if we replace the value of as x by 2 and then y as expressed here then we get this expression for the area to find the maximum area let us find the extremum values of a which is the point where d a d x or the derivative of a with respect to x vanishes so solving this equation for x we find that x is given by 56 .29 feet and then using this value of x in the expression for a y we find that y is equal to 28 .1 4, 5 feet.

02:01 We can also check that d2a, dx2 is negative, so this is indeed a maximum.

02:11 In the next problem, we are given a curve whose equation is equal to y by square root of x, and then a point whose coordinates are 9 comma 0.

02:19 Let us denote it by x0 y0 and let a point on the curve px1, y1, which is given by the x coordinate and the y coordinate equal to square root of x.

02:31 The distance of this point on the curve from this point 9 .0 is given by 9 minus x whole square plus square root of x minus 0 whole square so basically this gives us square root of 9 minus x whole square plus x now to find the closest point we have to extremize this value of d so let us take the derivative d d d x and then set it to zero to find the value of x we find x is equal to 17 by 2 equal to 8 .5 and then if we evaluate d2 d d x2 at this value of x we will find that this is indeed less than sorry this is indeed greater than zero which means that it is a minimum so that's the closest point next to we need to construct a metal box minimizing its cost so the faces or the top and the bottom parts of the metal box measures x by x it's a square and then its height is given by z so the top and the bottom is uh is constructed using a metal whose cost is three dollars per cubic the sides are constructed with another metal which costs $8 per square feet.

04:07 So the cost for the metal for the top and the bottom is equal to c1, that is equal to 3x square into 2 because we have two surfaces the top and the bottom.

04:17 And for the four sides, the cost is c2 which is given by 8 xz.

04:22 Exx is the area of each of this sides and there are four such sides.

04:27 Now the total volume of the metal box is given by v which is equal to x squared z and that is 50 cubic feet...

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