the next sums are given below-
1. A closed cylinder has total surface area equals to 600pi. Show that the volume , V cmcube, of this cylinder is given by the formula V= 300pi r- pi rcube, where r cm is the radius of the cylinder. Find the maximum volume of such a cylinder.
2. A sector of a circle has area 100cmsquare. Show that the perimeter of this sector is given by the formula
P=2r + 200/r , r> squareroot 100/pi. The radius of the sector is r cm.
Find the minimum value for the perimeter of such a sector.
3. A shape consists of a rectangular base with a semicircular top, as shown. Given that the perimeter of the shape is 40cm, show that its area, A cmsquare, is given by the formula A= 40r- 2rsquare- pi rsquare/2 where r cm is the radius of the semicircle. Find the maximum value for this area.
4. The shape shown is a wire frame in the form of a large rectangle split by parallel lengths of wire into 12 smaller equal-sized rectangles. Each smaller rectangle, the length is y mm and the width of that smaller rectangle is x mm. Given that the total length of wire used to complete the whole frame is 1512mm, show that the area of the whole shape is Ammsquare, where A = 1296x - 108xsquare/7. Find the maximum area which can be closed in this way.
5. The fixed point A has coordinates ( 8, -6, 5) and the variable point P has coordinate ( t, t, 2t).
a.) Show that APsquare= 6tsquare - 24t - 125.
b.) hence find the value of t for which the distance AP is least.
c.) Determine this least distance.
6. A wire is bent into the plane shape ABCDEA. Shape ABDE is a rectangle and BCD is a semicircle with diameter BD. The area of the region enclosed by the wire is R msquare, AE= x metres, AB= ED= y metres.The total length of the wire is 2 m.
a.) Find an expression for y in terms of x.
b.) Prove that R = x/8 (8 - 4x - pix)
Given that x can vary, using calculus and showing your working,
c.) find the maximum value of R. ( you do not have to prove that the value you obtain is a maximum)
7. A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1cm. The radii of the tin and the lid are both x cm. The tin and the lid are made from a thin sheet of metal of area 80pi cmsquare and there is no wastage. The volume of the tin is V cmcube.
a.) Show that V = pi ( 40x - xsquare - xcube).
Given that x can vary:
b.) Use differentiation to find the positive value of x for which V is stationary
c.) Prove that this value of x gives a maximum value of V.
d.) Find this maximum value of V.
e.) Determine the percentage of the sheet metal used in the lid when V is a maximum.
8. The part of the curve with equation y= 5- 1/2xsquare for which y> 0. The point P (x,y) lies on the curve and 0 is the origin.
a.) Show that OPsquare = 1/4xsquare - 4xsquare + 25.
Taking f(x)= 1/4xto the power of 4 - 4xsquare + 25.
b.) Find the values of x for which dy/dx = 0.
c.) hence , or otherwise, find the minimum distance from 0 to the curve, showing that your answer is a minimum.
9. The part of the curve with equation y= 3 + 5x + xsquare + xcube. The curve touches the x-axis at C. The points A and B are stationary points on the curve.
a. ) Show that C has coordinates (3, 0)
b. ) Using calculus and showing all your working, find the coordinates of A and B.
10. An open tank for storing water, ABCDEF. The sides ABFE and CDEF are rectangles. The triangular ends ADE and BCF are isosceles, and angle at AED is equal to angle at BFC which is 90 degree. The ends ADE and BCF are vertical and EF is horizontal.
Given that AD= x metres:
a.) Show that the area of triangle ADE is 1/4xsquare msquare.
Given also that the capacity of the container is 4000 mcube and that the total area of the two triangular and two rectangular sides of the container is S msquare.
b.) Show that S = xsquare/2 + 16000squareroot 2 / x
Given that x can vary:
c.) Use calculus to find the minimum value of S.
d.) Justify that the value of S you have found is a minimum.
There are 10 sums above and sir please each and every sum, I am expecting from u to show ur working nealty and in details and very clearful as well along with neatly diagrams.. Pls Pls Pls Pls.
Sir why these sums are said to be a toughest and difficult part than others?? is there any tips that I have to learn or know ?? do u have any tricky tips which helps to overcome the difficulties of the sums with different situation?? Sir those types of questions,the most difficult part is how to form a formula that they are given. I know there is a diagram// But using the diagram I have to make a formula, but to make a formula with different variables/terms, i found difficult// I dont understand in some formula why they put minus and plus within a formula and how it becomes minus and plus?
Pls i need ur best solution to this matter
i need ur reply asap.
thanx in advance
