Expert Help with High School Geometry

Show that angles of $\frac{\mathrm{\pi <!--\; \pi \; -->}}{4}$ and pi radians can both be trisected.

A rectangle has a perimeter of 36 inches. If the length of the rectangle is twice its width, then what is the area of the rectangle?

Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r ? 1). Let r = TVr+1 ? Vr ? 2 and Qr = Tr+1 ? Tr for r = 1, 2,

Marco came to a drawbridge that had a sign stating that the tolltaker would double his money each time he crossed the bridge, but Marco would have to paya fee of $1.20 per crossing. Marco decided he would cross the bridge 3 times, tosee how the system worked.
Sure enough, each time Marco crossed the bridge the toll takerdoubled the amount of money in Marco's pocket and then took his $1.20 toll charge. Marcowas surprised when , after the third crossing, he paid the $1.20fee and found himself with no money.

Write its converse. If the converse also true, combine the statements as a biconditional
If two segments have the same length, then they are congruent

What is the surface area of a cylindrical cylindrical ring, or toru, ifits out side diameter is 16mm and its inside diameter is 10mm?round off your answer to the nearnest whle number wich one is correct is it A 92mm or B 118mm or C 123mm or D 385 mm wich one is correct

a cylinder whose height is 8 inches has a volume of 128 pi cm3. if the radius is doubled and its height is cut in half, the volume of the resulting cylinder is...
please show how to do it i do not need the answer i know the answer is 256pi cm^3

Find the work done by a force F = 5i (magnitude 5 N) in moving an object along the line from the origin to the point (1,1) (distance in meters).

ÐABCand ÐPQR are supplementary angles and ÐABC is eleven timesas large as ÐPQR. Determine the measure of each angle.

A liquid vessel which is initially empty is in the form of an inverted regular hexagonal pyramid of altitude 25ft and base edge of 10ft. how much will the surface rise when 6,779 liters of water is added?

A cylindrical container has a base area 100m square and is 12m high if the container is one-third thid filled with water, what is volume of the water in the container? wich one is it is it A300m or B400m or C600 mor D1200m wich one is correct.

For a triangle ABC, prove that angle C is a right angle if and only if the median of side c is c/2

Find the slope of a line a. parallel and b. perpendicular to the line x+2y=2.

Show that lobachevsky's basic triangle formulas transform into standard formulas of spherical trigonometry if one replaces the sides a,b,c of the triangle by ia, ib, ic, respectively. Assume that the angle C is a right angle.
Formulas: sinAcot(productb)=sinBcot(producta), cosAcos(productb)cos(productc)...

Find the lateral surface area of a regular hexagonal pyramid whose edge measures 20 cm and the radius of a circle inscribed in the base is 9 squareroot of 3 cm.

Rectangle ABCD, shown here, will be a golden rectangle if AB/AD=BC/BE where AE = AD. Let AE = 12 and find the ratio of AB to AD. (Round your answer to three decimal places.)

Definition #1: an isosceles trapezoid is a trapezoid with exactly one pair of parallel sides and whose none parallel sides are congruent.
Definition #2: an isosceles trapezoid is a trapezoid with one pair of parallel sides and with the other pair of sides congruent.
a) Explain how isosceles trapezoid are related to rectangles under each Definition?
b) How are these Definitions related to the Definition of trapezoid ?
c) a quadrilateral has vertices at (-a,0), (a,0), (b,c) and (d,c). Under Definition #1, what conditions must be b and d satisfy if the quadrilateral is an isosceles trapezoid ? Under Definition #2, what conditions must be b and d satisfy if the quadrilateral is an isosceles trapezoid ?
d) Are the two Definitions equivalent? why or why not?

Calculate the volume of a regular pyramid of height h
Calculate the volume of a regular pyramid of height h, knowing that this pyramid is based on a convex polygon whose sum of inner angles is nπ and the ratio between the lateral surface and the base area is k.
Idea: Use the formula $\boxed{{\displaystyle \frac{lwh}{3}}}$ for finding the volume. But at first you have to Bash the problem to find the width and length and then after getting their values you have to plug it into the formula to find the volume.

Finding volume via integration
When we are asked to find the volume for example of a sphere, we slice it into numerous disks with small depth dr and get the infinitesimal volume dV. And then we sum those slices together by putting an integral symbol. But my question is the basic integration we learnt was to find area under the curve.In that case we proved that the area is simply the anti derivative of the function. But here how do we use integration by putting an integral symbol before Adr without proving that the derivative of the volume is the area of the disks? I am so disturbed by this as I didn't see any author giving the proof with proper intuition.

How to calculate the volume of cos(x) around the x-axis and the y-axis separately?
I have difficulties finding the right formula how to calculate the volume rotating cos(x) around the x-axis and y-axis separately can you please give me a hint how to do it?
The interval is $x=[-<!--\; -\; -->\frac{\mathrm{\pi <!--\; \pi \; -->}}{2},\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}]$