Math Answers

3. A trough having a trapezoidal cross section is full of water. If the trapezoid is 3ft wide at the top, 2ft wide at the bottom, and 2ft deep, find the total force owing to water pressure on one end of the trough.
4. Find the total force on the dam due to the fluid pressure:
A.Rectangle 200ft wide, 15ft high; water 10ft deep

1.An airplane at an altitude of 4400ft is flying horizontally away from an observer. At the instant when the angle of elevation is 45 degrees, the angle is decreasing at the rate of .05 rad/sec. How fast is the airplane flying at the instant?
2.A building 8ft high is 27/8ft from a building. Find the length of the shortest ladder which will clear the wall and rest with one end on the ground and the other end on the building. Also, find the angle which this ladder makes with the horizontal.
3.A man is walking along a sidewalk at the rate of 5ft/sec. A searchlight on the ground 30ft from the walk is kept trained on him. At what rate is the searchlight revolving when the man is 20ft away from the point on the sidewalk nearest the light?
4.A ladder 15ft long leans against a vertical wall. If the top slides down at 2ft/sec, how fast is the angle of elevation of the ladder decreasing, when the lower end is 12ft from the wall?

2.A right triangle has hypotenuse of length 13 and one leg of length 5. Find the dimensions of the rectangle of largest areas which has one side along the hypotenuse and the ends of the opposite side on the legs of this triangle.
3.A closed box with a square base is to have a volume of 2000 cu. Inches. The material for the top and bottom of the box is to cost Php3 per square inch, and the material, find the dimensions of the box.
1.Find the dimensions of the largest circle that can be inscribed in a square of 12 inches.
1.A manufacturer makes aluminum cups of a given volume ( 16 in3 ) in the form of right circular cylinders open at the top. Find the dimensions which use the least material.
2.Find the dimensions of the right circular cylinder of maximum volume which can be inscribed in a right circular cone of altitude 10 and radius 12.

I-ANALYTIC GEOMETRY/RECTILINEAR MOTION
1.Find the tangent line as directed to the curve Y = X4 + 2X3 – 2X2 -3X + 3 perpendicular to the line: X – 3Y = 2.
2.Find the tangent of the line as directed to the curve Y = X4 + 4X3 – 8X2 + 3X + 70 with slope 3.
3.Find the equation of the line tangent to the curve Y = 3X2 – 4X and parallel to the line X – 2Y + 6 = 0.
4.Find the equation of each normal line to the curve Y= X3 – 4X that is parallel to the line X + 8Y -8 = 0.
5.A particle moves along a straight line according to the law: S = 132 + 10t – 6t2 + 3t3. Find: a.) velocity and acceleration at any time t? b.) Velocity at t = 2 and c.) Acceleration at t= 3.
II-MAXIMA AND MINIMA
1.A box with a square base is to have an open top. The area of the material in the box is to be 100 in square. What should the dimensions be in order to make the volume as large as possible?

If y=e^ax 〖cos〗^(3 ) X 〖sin〗^2 X find dy/dx

Two roads intersect at point P at an angle of 120◦
, as shown in the figure. Car X is driving
from P towards A, and car Y is driving from P towards B. At a particular time, car X is 10
kilometers from P and traveling at 60 km/hr, while car Y is 12 kilometers from P and traveling
at 80 km/hr. How fast is the distance between the two cars changing?

I have the same question only with C(x)=72,000+40x and p(x)=300-x/20, 0≤x≤6000
(A) Find the maximum revenue.
​(B) Find the maximum​ profit, the production level that will realize the maximum​ profit, and the price the company should charge for each television set.
​(C) If the government decides to tax the company ​$5 for each set it​ produces, how many sets should the company manufacture each month to maximize its​ profit? What is the maximum​ profit? What should the company charge for each​ set?

2. For the function f(x) = 2x -arcsin x, a) locate all the local extrema and classify them
as maxima or minima. b) Are these local extrema also global extrema? Explain your answer.
4. For the curve y = ln(1+x^2), find a) intervals of increase and decrease and local extrema,
b) intervals of concavity and inection points, then use this information to c) sketch the graph.

3. Suppose f is an odd function which is differentiable everywhere and a is a positive number.
Show that for every a there exists c belong ( a; a) such that f'(c) = f(a)=a.

14. Two roads intersect at point P at an angle of 120, as shown in the figure. Car X is driving
from P towards A, and car Y is driving from P towards B. At a particular time, car X is 10
kilometers from P and traveling at 60 km/hr, while car Y is 12 kilometers from P and traveling
at 80 km/hr. How fast is the distance between the two cars changing?