# 2.2.294 Euler's method word problem

• MHB
• karush
In summary, George and his brother Bill make a paper boat and go out to play with it in the rain. The boat falls into a stream and is in danger of going into the sewer. The position of George and the boat are both at 0 at $t=0$. The velocity of the paper boat is modeled by the function $p(t) = [e^{0.05t}-\cos(0.35t)] \cdot \left(6-\dfrac{t}{5}\right)$. The estimated value of $p'(7)$ is 0.9153 meters per second.
karush
Gold Member
MHB
George makes a paper boat with his brother Bill and goes out in the rain to play with it. It falls in the stream along the curb, racing towards the sewer.
Let t be measured in seconds, p be the velocity of the paper boat in meters/second and g be George’s velocity, measured in meters/seconds.
At $t = 28$, if George has not retrieved his boat, it will fall in the sewer, Let the position of George and his boat both be 0 at $t = 0$
$$\begin{array}{rrrrrrr} x&0&7&14&21&28\\ g(t)&0&5&8&6&4 \end{array}$$
a. George's velocity is modeled by the function $g(t)$. Certain values of g are given in the table. Estimate the value of $g'(10.5)$, including units of measure.

the table points are on this Desmos plot. the first 3 pts are not a straight line but could represent a section of a parabola
then take the d/dx?

View attachment 9314

ok actually not sure what to do with this and its due tom

Last edited:
c. The velocity of the paper boat is modeled by the function
$\displaystyle p(t)=(e^{0.05t}-\cos(0.35t))\left[6-\dfrac{t}{5}\right]$
what is the value of $p'(7)$
$$p'=\frac{0.2\left(-t+30\right)\left(e^{0.05\left(t\right)}-\cos\left(0.35\left(t\right)\right)\right)}{t}$$
then
$$p'(7)=\frac{0.2\left(-t+30\right)\left(e^{0.05\left(t\right)}-\cos\left(0.35\left(t\right)\right)\right)}{t}=1.43868207031$$

Last edited:
$t=10.5$ is midway between $t=7$ and $t=14$

best estimate with the given data ...

$g'(10.5) \approx \dfrac{g(14)-g(7)}{14-7} = \dfrac{3}{7} \, m/s^2$

skeeter said:
$t=10.5$ is midway between $t=7$ and $t=14$

best estimate with the given data ...

$g'(10.5) \approx \dfrac{g(14)-g(7)}{14-7} = \dfrac{3}{7} \, m/s^2$
thanks

karush said:
c. The velocity of the paper boat is modeled by the function
$\displaystyle p(t)=(e^{0.05t}-\cos(0.35t))\left[6-\dfrac{t}{5}\right]$
what is the value of $p'(7)$
$$p'=\frac{0.2\left(-t+30\right)\left(e^{0.05\left(t\right)}-\cos\left(0.35\left(t\right)\right)\right)}{t}$$
then
$$p'(7)=\frac{0.2\left(-t+30\right)\left(e^{0.05\left(t\right)}-\cos\left(0.35\left(t\right)\right)\right)}{t}=1.43868207031$$

done

$p(t) = [e^{0.05t}-\cos(0.35t)] \cdot \left(6-\dfrac{t}{5}\right)$

$p'(t) = [e^{0.05t}-\cos(0.35t)] \cdot \left(-\dfrac{1}{5}\right) + \left(6-\dfrac{t}{5}\right) \cdot [0.05e^{0.05t}+0.35\sin(0.35t)]$

$p'(7) = [e^{0.35}-\cos(2.45)] \cdot \left(-\dfrac{1}{5}\right) + \left(\dfrac{23}{5}\right) \cdot [0.05e^{0.35}+0.35\sin(2.45)] = 0.9153$

## 1. What is Euler's method and how is it used in solving word problems?

Euler's method is a numerical method for approximating the solution to a differential equation. It is used in solving word problems by breaking down the problem into smaller steps and using a simple formula to calculate the approximate solution at each step.

## 2. Can Euler's method be used for any type of differential equation?

No, Euler's method is best suited for simple, first-order differential equations. It may not provide accurate results for more complex equations or those with multiple variables.

## 3. How is Euler's method different from other numerical methods for solving differential equations?

Euler's method is a first-order method, meaning that it only uses information from the current step to calculate the next step. Other methods, such as the Runge-Kutta method, use information from multiple steps to improve accuracy.

## 4. Are there any limitations or drawbacks to using Euler's method?

Yes, Euler's method can be less accurate than other methods and may not provide a precise solution for more complex problems. It also requires a smaller step size to achieve more accurate results, which can be time-consuming.

## 5. Can Euler's method be used for real-world applications?

Yes, Euler's method can be used in various fields, such as physics, engineering, and economics, to approximate solutions to differential equations. However, it should be used with caution and its limitations should be considered when applying it to real-world problems.

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