- #1
ineedhelpnow
- 651
- 0
View attachment 3931
Hi!
I'm stuck
I was thinking I need to use the equation $f=kw^2$ but I'm really not sure.
Hi!
I'm stuck I was thinking I need to use the equation $f=kw^2$ but I'm really not sure.
Growth proportional to square root of worth.
- MHB
- Thread starter ineedhelpnow
- Start date
In summary, the problem involves a firm with a fortune that grows at a yearly rate proportional to the square root of its worth. This can be represented by the differential equation \frac{dW}{dt}=k\sqrt{W}, where W is the firm's worth and t is time. Setting up an initial value problem, we can solve for the function W(t) and determine the constant k.
- #2
Ackbach
Gold Member
MHB
- 4,155
- 89
So that first sentence there: "A firm has a fortune that grows at a yearly rate proportional to the square root of its worth." Can you translate that sentence into a differential equation?
- #3
SuperSonic4
Gold Member
MHB
- 190
- 0
ineedhelpnow said:Hi!
I was thinking I need to use the equation $f=kw^2$ but I'm really not sure.
A firm has a fortune that grows at a yearly rate proportional to the square root of its worth
Suppose the firm's fortune is given by \(\displaystyle n(w)\), the firm's worth is \(\displaystyle w\) and the time taken is \(\displaystyle t\). Can you set up an differential equation to show the firm's fortune in terms of worth at a given time
\(\displaystyle \dfrac{dn}{dt} \propto \sqrt{w} \ \rightarrow \dfrac{dn}{dt} = k \sqrt{w}\) where \(\displaystyle k\) is some constant
edit: making it clear that \(\displaystyle n\) is a function of \(\displaystyle w\)
Last edited:
- #4
ineedhelpnow
- 651
- 0
Sorry for the late response. The assignment was due and I got really distracted after but I'd still like to work through the problem to see how it's done.
@Ackbach No, I don't. Well until SuperSonic showed it. :p I still don't understand where to go from there. I'm super lost.
@SuperSonic Thank you :D
@Ackbach No, I don't. Well until SuperSonic showed it. :p I still don't understand where to go from there. I'm super lost.
@SuperSonic Thank you :D
- #5
MarkFL
Gold Member
MHB
- 13,288
- 12
I would let time $t=0$ be at 1/2 year ago, and let $W$ be measured in millions of dollars. Then I would state the IVP:
\(\displaystyle \frac{dW}{dt}=k\sqrt{W}\) where \(\displaystyle W(0)=1,\,W\left(\frac{1}{2}\right)=4\)
Separating variables and using the boundaries as limits, we have:
\(\displaystyle \int_1^{W} u^{-\frac{1}{2}}\,du=k\int_0^t\,dv\)
\(\displaystyle 2\left[u^{\frac{1}{2}}\right]_1^{W}=k[v]_0^t\)
\(\displaystyle 2\left(\sqrt{W}-1\right)=kt\)
Since $k$ is a constant, we can divide through by 2 and still have a constant:
\(\displaystyle \sqrt{W}-1=kt\)
\(\displaystyle W(t)=(kt+1)^2\)
Now, using the second point to determine $k$
\(\displaystyle \left(\frac{k}{2}+1\right)^2=4\)
\(\displaystyle \frac{k}{2}+1=2\)
\(\displaystyle k=2\)
Hence:
\(\displaystyle W(t)=(2t+1)^2\)
\(\displaystyle \frac{dW}{dt}=k\sqrt{W}\) where \(\displaystyle W(0)=1,\,W\left(\frac{1}{2}\right)=4\)
Separating variables and using the boundaries as limits, we have:
\(\displaystyle \int_1^{W} u^{-\frac{1}{2}}\,du=k\int_0^t\,dv\)
\(\displaystyle 2\left[u^{\frac{1}{2}}\right]_1^{W}=k[v]_0^t\)
\(\displaystyle 2\left(\sqrt{W}-1\right)=kt\)
Since $k$ is a constant, we can divide through by 2 and still have a constant:
\(\displaystyle \sqrt{W}-1=kt\)
\(\displaystyle W(t)=(kt+1)^2\)
Now, using the second point to determine $k$
\(\displaystyle \left(\frac{k}{2}+1\right)^2=4\)
\(\displaystyle \frac{k}{2}+1=2\)
\(\displaystyle k=2\)
Hence:
\(\displaystyle W(t)=(2t+1)^2\)
1. What does "growth proportional to square root of worth" mean?
This phrase refers to a mathematical relationship between two variables, where one variable (growth) is directly proportional to the square root of the other variable (worth). This means that as the worth increases, the growth also increases, but at a slower rate.
2. How is this concept applicable in science?
This concept is applicable in many areas of science, such as population growth, economic growth, and biological growth. It can also be used to describe the relationship between variables in experiments or studies.
3. What are the implications of this relationship?
The implications of this relationship are that as the worth of something increases, the growth will not increase at a linear rate. This can have an impact on predictions and projections, as well as decision making.
4. Can you provide an example of "growth proportional to square root of worth"?
One example could be the growth of a company's profits. As the company's worth (measured by assets or market value) increases, the profits may also increase, but at a slower rate. Another example could be the growth of a population. As the population increases, the rate of growth may slow down due to limited resources.
5. How does this concept compare to other growth relationships?
This concept differs from other growth relationships, such as linear or exponential growth, in that the growth rate is not constant. It also differs from inverse relationships, where one variable decreases as the other increases. The relationship between growth and square root of worth is unique in that it shows a slower growth rate as the worth increases.
Similar threads
-
Differential Equations
-
Quantum Physics
-
Introductory Physics Homework Help
-
Differential Equations
-
General Math
-
Introductory Physics Homework Help
-
General Math
-
Precalculus Mathematics Homework Help