0
votes
1answer
20 views

Maximize area of a rectangle between parabola and a line

I was given a task to maximize the area of a rectangle that can be inscribed between parabola $y=1-x^2$ and a line $y=0$ such that one side of the rectangle lies on the $x$ axis. My idea is to somehow ...
1
vote
3answers
122 views

Max perimeter of triangle inscribed in a circle

What is the maximum perimeter of a triangle inscibed in a circle of radius $1$? I can't seem to find a proper equation to calculate the derivative.
0
votes
1answer
89 views

Finding Arc Length using First Fundamental Form

Let $v=ln(u+\sqrt{u^2+1})+C$ be the curve given on the right helicoid $x=u\cos(v),y=u\sin(v),z=2v$. Calculate the arc lengths of this curve between the points $M_1(0,0)$ and $M_2(1,ln(1+\sqrt{2}))$ ...
2
votes
1answer
57 views

How do I determine between positive and negative inflection

Is it possible to identify whether an inflection point such as this example, contained in y = x^3 from the wikipedia: Is positive or negatively oriented (i.e. the ...
1
vote
1answer
25 views

How to maximize the function

I have a triangle $T=ABC$. I want to calculate $\max (a-b)$, where the the angle $ABC = \beta$, and $|AB|=c$ is fixed (pre-known). My guess is $c\times\cos (\beta)$, but I want to prove it. Let ...
0
votes
1answer
148 views

Area of a folded paper

One corner of a rectangular sheet of paper is folded over so as to reach the opposite edge(lengthwise) of the sheet. If area of the folded paper is minimum, show the crease divided the width in 2:3. ...
2
votes
2answers
147 views

Cheapest can problem

A cylindrical can which must hold 1000 mL is set to be designed so the least amount of material is necessary to make the can. What should the radius be? What is the height of the can? What is the ...
4
votes
2answers
124 views

Minimizing the length of a pipeline between cities

I have been trying to minimize piping going to two different cities. City A is located at $(0,4)$ and city B is located at $(6,3)$. The cities must connect to the $x$-axis (the main pipe line.) It ...
1
vote
3answers
167 views

Trigonometry tangent line question

How would I figure this out. Find all x values between $0$ and $2\pi$ where the line tangent to the graph of $y=\frac{\cos x}{2+ \sin(x)}$ is horizontal. I did the deriavative ...
3
votes
3answers
124 views

Calculus word problem dealing with rectangle

How would I figure this problem out. A rectangle has a base B and a height H. Assuming that is area remains constant express the rate of change of the height with respect to the base. this is what I ...
1
vote
2answers
1k views

Surface area of a sphere?

How would I solve the following problem? Find the rate of change of surface area of a sphere with respect to its diameter D. I know the formula for surface area of a sphere is $A=4\pi r^2$ So I ...
0
votes
1answer
151 views

Points on curve where tangent are equally inclined

What are the points on the curve $ x^{3/2} + y^{3/2} = a^{3/2} $ where the tangents are equally inclined to the axes?
2
votes
1answer
39 views

What is the result of deriving a polygon?

If you define a polygon - say, for simplicity, a triangle - as a list of functions, defined piecewise, for example: $a(x)=2x$ defined on $[0,1]$ $b(x)=-2x+2$ defined on $[1,2]$ $c(x)=0$ defined on ...
1
vote
1answer
73 views

Finding the angle of a by-pass road given the lengths of two roads forming a right angle

The north-east section of a municipality is a rectangle whose 17 km west boundary is Highway 7 and whose 20 km south boundary is Highway 15. A by-pass road is planned to alleviate traffic congestion. ...
1
vote
2answers
4k views

What is the maximum volume of a cylinder that can fit in a sphere of a constant radius?

The first question that comes into my mind here is whether any cylinder that touches(at 4 pts) the circumference of the sphere and does not go out of it, has equal volume? Second, how do i ...
1
vote
1answer
235 views

Rectangular box of largest surface area inscribed in a sphere of radius 1?

I am having trouble trying to follow a textbook example of such a problem. Using the Lagrange's Multiplier method, we could set up a set of equations like below: $f(x,y,z)=8(xy+yz+zx)$ ...
0
votes
1answer
136 views

Directional derivative (muiltivariable calculus)

I have an encountered an example in my text book which I don't fully see the intuition of. I will write out the part of the example I'm struggling with: A hiker is standing beside a stream on the ...
3
votes
3answers
227 views

Help me name or find the existing name for this geometric concept!

This may have a proper name, if so - let's discuss. If not, let's name it. This is for a web application in C#, so whatever we call it I will start naming as such in my code. I'm taking GPS data as a ...
0
votes
1answer
154 views

Chain Rule and Homogenous Coordinates

I have a vector $\tilde{p} = (x,y,z)$ (homogenous coordinates). The corresponding non-homogenous vector is $p = (x/z, y/z)$. Now the $\tilde{p}$ is a result of some linear transform $R(\theta)$ of ...
3
votes
1answer
373 views

Maximum triangle area

I have a small problem. Consider I have a triangle. Which maximum area can it cover if two of his medians are 3 and 8? I think I'll need to use derivative here, but firstly I need to find a function ...