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Calculus
Use a triple integral to determine the volume of the region bounded by z =
p
x
2 + y
2 and z = x
2 + y
2
In 1st octant
Calculus
from the top of a 60m tall lighthouse, the angle of depression of a ship moving east away from the lighthouse at 8kmph is 15 degrees. Calculate the angle of depression of the ship from the top of the lighthouse two minutes later, given that the ship stays on the same course at the same speed.
Calculus
P(2, 3) lies on the curve y=(1/2)x^2+1. O is the origin, M is the foot of the perpendicular to the x-axis and the curve intersects the y-axis at A. The area bounded by AO, OM, MP and the curve PA of the curve is rotated through four right angles about the y-axis. Find the volume of the solid formed, giving your answer as a multiple of π.
Calculus
The curve y = 1 -(1/4)x^2 intersects the positive side of the x-axis at A and the y-axis at B. O is the origin. Calculate the volume generated when the finite area bounded by BO, OA, and the arc AB is rotated through four right angles
i. about the x-axis
ii. about the y-axis.
Give each answer as a multiple of π.
i. about the x-axis
ii. about the y-axis.
Give each answer as a multiple of π.
Calculus
The straight line y = 3x − 3 intersects the parabola y^2 = 12x at the points P and
Q. Show that P and Q lie on opposite side of the x-axis and calculate the finite
area bounded by the chord and the arc PQ of the parabola.
Q. Show that P and Q lie on opposite side of the x-axis and calculate the finite
area bounded by the chord and the arc PQ of the parabola.
Calculus
The curve y = (1 + x)(3 − x) intersects the positive side of the x-axis at A. The
tangent to the curve at O, the origin, intersects the curve again at P. Calculate
i. the finite area bounded by the curve and OA,
ii. the finite area bounded by the curve and OP
tangent to the curve at O, the origin, intersects the curve again at P. Calculate
i. the finite area bounded by the curve and OA,
ii. the finite area bounded by the curve and OP
Calculus
“Count Every Particle” is a game conducted by the teachers and students to generate funds for the victims of Typhoon. The game was, the participant will buy a ticket so that they have chance to guess the exact number of particles of the sand that is placed in a vase. The following hints were given:
(1) It is assumed that the volume of every particle is 6.4π x 10^-12 cm^3.
(2) The height of the vase is 32 cm; and
(3) The equation of the curve that the vase made of is given by y=x^2 - 6 wit domain x≥ -2.
To win this game, calculate the exact number of particles that can be placed in the vase.
(1) It is assumed that the volume of every particle is 6.4π x 10^-12 cm^3.
(2) The height of the vase is 32 cm; and
(3) The equation of the curve that the vase made of is given by y=x^2 - 6 wit domain x≥ -2.
To win this game, calculate the exact number of particles that can be placed in the vase.
Calculus
Supposes a flexible cable is suspended between two towers that are 200 feet apart forms a curve whose equation is y=75(e^x/150 + e^-x/150). Calculate the length of the cable.
Calculus
A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is: (a) maximum? (b) minimum?
Calculus
e {𝑎𝑛 }𝑛=1 ∞ is defined by 𝑎𝑛 = 2𝑛−3 3𝑛+4 for 𝑛 ∈ ℕ Prove that {𝑎𝑛 }𝑛=1 ∞ is a bounded sequence. iii) Find lim𝑛→∞ 𝑎�
