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Consider the problem of a sphere of material that starts at a non-uniform temperature, $T = r^{2}$ and is covered with insulation on the outer surface so that no heat gets out. We take the coordinate $r$ to measure position radially from the centre of the sphere with the outer surface given at $ r = 1.$ We take t as time and the variable $ T (r, t)$ as the temperature. The equation governing the heat flow, is the heat equation $$\frac{\delta T}{\delta t}=\frac{1}{r^2}\frac{\delta}{\delta r}(r^2\frac{\delta T}{\delta r}), 0\leq r\leq 1, t>0$$ We insist that T remains finite as $ r → 0$ and the outer surface boundary condition is $$\frac{\delta T}{\delta t}(1,t)=0, t>0$$ with the initial condition $$T(r,0) = r^2, 0 < r < 1.$$ (a) Find the solution T (t, r) using separation of variables. Note: you can leave the answer in a form where eigenvalues are given by the roots of an equation. By exploiting the orthogonality of the eigenfunctions you should give the integral formulas necessary to compute the coefficients in the solutions. You do not need to evaluate the integrals. (b) Find numerically, to five decimal places accuracy, the value of the smallest of the eigenvalues in (a), corresponding to the first non-constant term in the solution. This is what I have obtained so far: $$\frac{\delta T}{\delta t}=\frac{1}{r^2}\frac{\delta}{\delta r}(r^2\frac{\delta T}{\delta r})$$ $=\frac{\delta^2 T}{\delta r^2}+ \frac{2}{r} \frac{\delta T}{\delta r}$ Let $T=R(r)Q(t)$ $\frac{\delta T}{\delta t}= R(r)Q'(t)$ $\frac{\delta T}{\delta r}= R'(r)Q(t)$ $\frac{\delta^2 T}{\delta r^2}= R''(r)Q(t)$ Then putting back into the equation we get $$\frac{Q'(t)}{Q(t)}=\frac{R''(r)}{R(r)} + \frac{2}{r}\frac{R'(r)}{R(r)}= \mu$$ which gives the two equations: $$Q'(t)-\mu Q(t)=0$$ and $$rR''(r)+2R'(r)-\mu rR(r)=0$$ Then I have done Let $o(r)=rR(r)$ $o'(r)=R(r)+rR'(r)$ $o''(r)=2R'(r)+rR''(r)$ Putting this back into the equation we get $o''(r)-\mu o(r)=0$ Now how do I proceed? using the boundary conditions how do I implement this into the equation.

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The problem I was given: Calculate the value of the following determinant: $\left \begin{array}{ccc} \alpha & 1 & \alpha^2 & -\alpha\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 & 2\alpha & 2\alpha\\ 1 & 1 & \alpha^2 & -\alpha \end{array} \right $ For which values of $\alpha \in \mathbb R$ is the following system of linear equations solvable? $\begin{array}{lcl} \alpha x_1 & + & x_2 & + & \alpha^2 x_3 & = & -\alpha\\ x_1 & + & \alpha x_2 & + & x_3 & = & 1\\ x_1 & + & \alpha^2 x_2 & + & 2\alpha x_3 & = & 2\alpha\\ x_1 & + & x_2 & + & \alpha^2 x_3 & = & -\alpha\\ \end{array}$ I got as far as finding the determinant, and then I got stuck. So I solve the determinant like this: $\left \begin{array}{ccc} \alpha & 1 & \alpha^2 & -\alpha\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 & 2\alpha & 2\alpha\\ 1 & 1 & \alpha^2 & -\alpha \end{array} \right $ = $\left \begin{array}{ccc} \alpha - 1 & 0 & 0 & 0\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 & 2\alpha & 2\alpha\\ 1 & 1 & \alpha^2 & -\alpha \end{array} \right $ = $(\alpha - 1)\left \begin{array}{ccc} \alpha & 1 & 1\\ \alpha^2 & 2\alpha & 2\alpha \\ 1 & \alpha^2 & -\alpha \end{array} \right $ = $(\alpha - 1)\left \begin{array}{ccc} \alpha & 1 & 0\\ \alpha^2 & 2\alpha & 0 \\ 1 & \alpha^2 & -\alpha - \alpha^2 \end{array} \right $ = $-\alpha^3(\alpha - 1) (1 + \alpha)$ However, now I haven't got a clue on solving the system of linear equations... It's got to do with the fact that the equations look like the determinant I calculated before, but I don't know how to connect those two. Thanks in advance for any help. (:

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i. Use MS Excel Data Analysis ToolPak to perform a multiple regression analysis using Quality as the response variable and Helpfulness and Clarity as the explanatory variables. Write down the corresponding coefficient estimates and provide the regression output. j. Perform an F-test for the overall usefulness of the model in part i) using a 5% significance level. Make sure you follow all the steps for hypothesis testing indicated in the Instructions section and clearly state your conclusion. k. Test manually if the Clarity variable is significant in the model in part i). Make sure you follow all the steps for hypothesis testing indicated in the Instructions section and clearly state your conclusion. l. Using the adjusted R2 criterion, does including Clarity as an additional predictor variable improve the model in part i)? Explain why it is better to use the adjusted R2 over the R2 to determine if the addition of this new variable improves the model. Regression Statistics ANOVA Multiple R 0.998544859 df SS MS F Significance F R Square 0.997091836 Regression 2 255.2639136 127.6319568 62229.00058 0 Adjusted R Square 0.997075813 Residual 363 0.744514614 0.002051004 Standard Error 0.045288017 Total 365 256.0084282 Observations 366 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -0.020353502 0.010520921 -1.934574223 0.0538193 -0.04104311 0.000336106 -0.04104311 0.000336106 helpfulness 0.538358378 0.007216008 74.60611907 2.8925E-222 0.524167949 0.552548808 0.524167949 0.552548808 clarity 0.465505241 0.00707634 65.78333849 8.6445E-204 0.451589474 0.479421009 0.451589474 0.479421009

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  QUESTION 1 On a particular day, Mobay Electronics produced 10000 electric bulbs at a mean production rate of 10 hours. Assume that the known population standard deviation is 2000 hours   a. Why do we need a confidence interval                                                                         We need to construct a confidence interval because people may take samples from a population and end up with different result. Therefore, we do not know the true value until we create a confident interval   b. Construct a 95% confidence interval for the mean production time of the light bulbs by Mobay electronics      QUESTION 2 Given the following numbers 2, 2, 3, 2, 4, 8, 0 Find a. The mean                                                                            b. the mode                                                                          c. the median                                                       d. the variance                                                    QUESTION 3 The average weight of 20 students in a certain school was found to be 165lbs with a standard deviation of 4.5. Construct a 95% confidence interval for the population mean. Please show all working  to show how you arrived at the answer QUESTION 4 At a large restaurant, 3 out of 5 customers ask for water with their meal. A random sample of 10 customers is selected. Find the probability that a. Exactly 6 ask for water with their meal                      b. Less than 9 ask for water with their meal              QUESTION 5 Patients arrive at a hospital Accident and Emergency department at random at a rate of 6 per hour. Find the probability that during any 90 minute period, the number of patients arriving at the hospital Accident and Emergency department is Exactly 7                                                                                            At least 1                                                                                                                                                                                                                                                                                                                                                                    

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