Find the curvature K of the curve at the point r(t)=e^tcos tcdot i+e^tsin tcdot j+e^tk,P(1,0,1)

Upper Level Math
asked 2020-12-28
Find the curvature K of the curve at the point \(r(t)=e^t\cos t\cdot i+e^t\sin t\cdot j+e^tk,P(1,0,1)\)

Answers (1)

\(r(t)=e^t\cos t\cdot i+e^t\sin t\cdot j+e^tk\)
\(r'(t)=e^t(\cos t-\sin t)i+e^t(\sin t+\cos t)j+e^tk\)
Find Magnitude
\(|r'(t)|=e^t\sqrt{(\cos t-\sin t)^2+(\sin t+\cos t)^2+1}\)
\(|r'(t)|=e^t\sqrt{(\cos^2t-\sin^2t-2\cos t\sin t)^2+(\sin^2t+\cos^2t+2\cos t\sin t)^2+1}\)
Recall that: \(T(t)=\frac{r'(t)}{|r'(t)|}\)
Therefore \(T(t)=\frac{(e^t(\cos t-\sin t)i+e^t(\sin t+\cos t)j+e^tk)}{(e^t\sqrt3)}\)
\(T(t)=\frac{1}{\sqrt3[(\cos t-\sin t)i+(\sin t+\cos t)j+k]}\)
Differentiate \(T'(t)=\frac{1}{\sqrt3[(-\sin t-\cos t)i+(\cos t-\sin t)j+0k]}\)
Find Magnitude \(|T'(t)|=\frac{1}{\sqrt3\sqrt{(-\sin t-\cos t)^2+(\cos t-\sin t)^2+0)}}\)
\(|T'(t)|=\frac{1}{\sqrt3\sqrt{(\sin^2t+\cos^2t+2\sin t\cos t)+(\cos^2t+\sin^2t-2\cos t\sin t)+0)}}\)
Recall that \(k(t)=\frac{\frac{\sqrt2}{\sqrt3}}{(e^t\sqrt3)}=\frac{\sqrt2}{(3e^t)}\)

Relevant Questions

asked 2021-01-02
The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. Then the volume (in centimeters cubed) of a ball of radius r centimeters is \(\displaystyle\frac{{4}}{{3}}\pi{r}^{{3}}\). The surface area is \(\displaystyle{4}\pi{r}^{{2}}\).Set up the differential equation for how r is changing. Then, suppose that at time t = 0 minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters. At what time t will the snowball be completely melted.
asked 2021-03-08
In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of j - k?
asked 2021-01-02
Find a transformation from u,v space to x,y,z space that takes the triangle \(\displaystyle{U}={\left[\begin{array}{cc} {0}&{0}\\{1}&{0}\\{0}&{1}\end{array}\right]}\) to the triangle
asked 2021-02-27
a) To graph: the \({k}{e}{r}{\left({A}\right)},{\left({k}{e}{r}{A}\right)}^{\bot}{\quad\text{and}\quad}{i}{m}{\left({A}^{T}\right)}\)
b) To find: the relationship between im \((A^{T})\) and ker (A).
c) To find: the relationship between ker(A) and solution set S
d) To find vecx_0 at the intersection of \({k}{e}{r}{\left({A}\right)}{\quad\text{and}\quad}{\left({k}{e}{r}{A}\right)}^{\bot}\)
e) To find: the lengths of \(\vec{{x}}_{{0}}\) compared to the other vectors in S
asked 2020-11-16
Advanced Math
A bipolar alkaline water electrolyzer stack module comprises 160 electrolytic cells that have an effective cell area of \(\displaystyle{2}{m}^{{2}}\). At nominal operation, the current density for a single cell of the electrolyzer stack is 0.40 \(\displaystyle\frac{{A}}{{c}}{m}^{{2}}\). The nominal operating temperature of the water electrolyzer stack is \(\displaystyle{70}^{\circ}\) C and pressure 1 bar. The voltage over a single electrolytic cell is 1.96 V at nominal load and 1.78 V at 50% of nominal load. The Faraday efficiency of the water electrolyzer stack is 95% at nominal current density, but at 50% of nominal load, the Faraday efficiency decreases to 80%.
(Give your answer to at least three significant digits.)
Calculate the nominal stack voltage:
Answer in V
Calculate the nominal stack current:
Answer in A
Calculate the nominal power on the water electrolyzer stack:
Answer in kW
asked 2021-01-16
Advanced Math
Find the final form of the median and range for uniform distribution (0,1) for order statistics.
asked 2020-12-31
Advanced Math
Find the final form of the median and range for uniform distribution (0,1) for order statistics.
asked 2021-02-08
Tories bookcase holds one algebra book one geometry book and 12 books about advanced engineering mathematics. Books are taken at random from her book case, one after another without replacement until an engineer book is had, at which no more books are taken. The random variable Z represents the total number of books taken during this experiment.
Find \(\displaystyle{P}{r}{\left[{z}={3}\right]}\)
asked 2021-02-02
A payment of 1500$ is due in three months.Find the equivalent value at nine months if the interest rate is 4%
select one
asked 2021-01-24
A bird flies in the xy-plane with a position vector given by \(\displaystyle\vec{{{r}}}={\left(\alpha{t}-\beta{t}^{{3}}\right)}\hat{{{i}}}+\gamma{t}^{{2}}\hat{{{j}}}\), with \(\displaystyle\alpha={2.4}\ \frac{{m}}{{s}},\beta={1.6}\ \frac{{m}}{{s}^{{3}}}\) and \(\displaystyle\gamma={4.0}\ \frac{{m}}{{s}^{{2}}}\). The positive y-direction is vertically upward. At the bird is at the origin.
Calculate the velocity vector of the bird as a function of time.
Calculate the acceleration vector of the bird as a function oftime.
What is the bird's altitude(y-coordinate) as it flies over x=0 for the first time after ?