Mathematics Answer Note #63

Question 1 The temperature in Alaska drops steadily from 10oC as winter approaches by 0.5 C every hour from midday through to midnight. a) Is this relationship linear? b) Use the graph to estimate the temperature at 8 pm. https://www.elearnoncloud.com/Storage/eldata/DicFiles/Canada/ONTG9F3_E/images/Q n4.jpg Answer Based on the given information and graph, we can observe that the temperature in Alaska follows a decreasing trend from 10oC at midday to 0oC at midnight, with a decrease of 0.5oC per hour. This leads us to the question of whether this relationship is linear. To answer part (a) of the question, we must first understand what a linear relationship is. A linear relationship is a relationship between two variables where the change in one variable is directly proportional to the change in the other variable. In simpler terms, a linear relationship is one that can be plotted on a straight line. To determine if the relationship between the temperature in Alaska and time is linear, we can plot the data points from the given graph on a scatter plot and observe if they form a straight line. Upon plotting the data, we can see that the data points form a straight line, indicating a linear relationship between the temperature and time. To estimate the temperature at 8 pm, we can use the formula for a linear relationship, y = mx + b, where y is the dependent variable (temperature in this case), m is the constant rate of change (0.5oC per hour), x is the independent variable (time in this case), and b is the y-intercept (10oC in this case). Substituting the values, we get: y = 0.5x + 10 To estimate the temperature at 8 pm, we need to substitute 8 pm (20:00) as the value for x. Therefore, the estimated temperature at 8 pm would be: y = 0.5(20) + 10 = 20.5 oC Hence, we can estimate that the temperature at 8 pm in Alaska would be approximately 20.5oC. Useful Sources: 1. "Linear Equations." Khan Academy, https://www.khanacademy.org/math/algebrahome/alg-eq-linear/alg-eq-linear-basics/v/solving-basic-linear-equations. Accessed 10 November 2021. 2. "Slope-Intercept Formula." Math is Fun, https://www.mathsisfun.com/equation_of_line.html. Accessed 10 November 2021. In conclusion, the relationship between the temperature in Alaska and time is linear, and the estimated temperature at 8 pm is 20.5oC. This can be calculated using the formula for linear equations and the data points from the given graph. Further research on linear relationships and equations may provide a better understanding of the concept. Question 2 Write a Explicit Function for the following problem claire has 300$ in an amount .she decides she gonna take out half of whats left in there at the end of each month Answer To create a function for this problem, we need to first understand the variables involved. The initial amount of money, $300, will be represented by the variable 'P'. The amount taken out at the end of each month, which is half of what is left, will be represented by the variable 'T'. Finally, the number of months will be represented by the variable 'n'. Thus, the function can be written as: P(n+1) = (P(n) - T) / 2 To break this function down further, P(n) represents the amount of money at the beginning of the nth month, and P(n+1) represents the amount of money at the beginning of the (n+1)th month. So, for example, P(1) would represent the initial amount of $300, while P(2) would represent the amount at the beginning of the second month, after the first month's withdrawal. To solve this function, we can use a table and plug in values for 'n'. Month (n) | Amount at beginning of month (P(n)) | Amount taken out (T) | Amount left at end of month 1 | $300 | $0 | $300 2 | $150 | $75 | $75 3 | $37.50 | $18.75 | $18.75 4 | $9.38 | $4.69 | $4.69 5 | $2.34 | $1.17 | $1.17 We can see that the amount left at the end of each month is half of what is left from the previous month. This is because the function is taking the previous amount and subtracting half of it (half of what's left), so each time the amount is getting divided by 2. Question 3 Letx be the number of newspaper ads, y be the number of radio ads, and z be the number of TV ads. Step 2: The total cost of advertising is 200x + 100y + 800z Step 3: The total number of women who Answer In order to accurately answer this question, we first need to understand the context and variables given. The given variables represent the number of advertisements in three different forms (newspaper, radio, and TV), while the equation (200x + 100y + 800z) calculates the total cost of all these advertisements. However, the question does not specify the purpose or target audience for these advertisements. If we assume that the intention of these advertisements is to reach out to women, we will need to gather more information in order to answer the question. Some important factors to consider would be the target demographics and locations for the advertisements, the effectiveness and reach of each form of advertisement, and the potential impact on the target audience. To further break down the given equation, we can see that the cost of advertising on newspapers (200x) is twice that of radio (100y) and eight times that of TV (800z). This could be due to the reach and effectiveness of newspaper advertisements, which tend to have a wider audience compared to radio or TV ads. Next, we can also calculate the average cost per advertisement by dividing the total cost by the total number of ads. For example, if we have a total of 100 newspaper ads (x = 100), 50 radio ads (y = 50), and 20 TV ads (z = 20), the total cost would be 200(100) + 100(50) + 800(20) = $32,000. To find the average cost per advertisement, we divide $32,000 by the total number of ads (100 + 50 + 20), resulting in a cost of $256 per advertisement. Furthermore, if the target audience for these advertisements is women, we could also consider breaking down the cost and reach of each form of advertisement specifically for this demographic. This could provide further insights into the effectiveness of each form of advertisement for the target audience. Question 4 3. It is shown that in the next 5 years the price of a Toyota car will vary with time, If P (t) = 150 - 10t + 7.5t ^ 2 where t is time in years and P(t) is the price in Naira. The number of Toyota cars sold yearly by the company is expected follow: where q(t) is the number of cars sold and t is time in years: q(t) = 2800 + 75t - 10t ^ 2 (1) Find R(t) and R_{2} ^ 2 * (t) (II) Compute R(interpret your result R(2) and interpret your result (iv) Compute R(4) and interpret your result Answer According to the given information, the price of a Toyota car will vary with time and can be represented by the equation P(t) = 150 - 10t + 7.5t^2 where t is time in years and P(t) is the price in Naira. This means that the price of a Toyota car will change over the next 5 years and can be determined by substituting different values of t into the equation. To determine the number of Toyota cars sold yearly by the company, we will use the equation q(t) = 2800 + 75t - 10t^2, where q(t) is the number of cars sold and t is time in years. This equation shows that the number of cars sold will also vary over the next 5 years and can be calculated by substituting different values of t. Now, we need to find the revenue function R(t) and the coefficient of determination R^2(t) for the number of Toyota cars sold yearly. The revenue function is given by R(t) = q(t) x P(t), which means it is the product of the number of cars sold and the price of each car. Thus, we have R(t) = (2800 + 75t - 10t^2)(150 - 10t + 7.5t^2). Simplifying this equation gives us R(t) = -7.5t^3 + 1725t^2 - 50t + 420000. This is the revenue function in terms of time. Next, we can calculate R^2(t) by using the formula R^2(t) = 1 - SSres / SStot, where SSres is the residual sum of squares and SStot is the total sum of squares. To calculate SSres, we first need to find the predicted values of R(t) using the revenue function. We will substitute the values of t (0, 1, 2, 3, 4, 5) into the function to get the predicted revenue values for each year. The actual values of R(t) can be calculated by multiplying the number of cars sold, q(t), with the price of each car, P(t). Once we have the actual and predicted values, we can use the formula to calculate SSres and then R^2(t). Interpreting the result of R^2(t), we will get a value between 0 and 1. A value closer to 1 means that there is a stronger relationship between the number of cars sold and the price, indicating that the revenue function is a good fit for the data. In this case, a higher R^2(t) value would mean that the revenue function is a good predictor for the company's sales in the next 5 years. To further understand the relationship between time and the number of cars sold, we can compute R(2) and R(4) which will give us the predicted revenue for the company at the end of the second and fourth year, respectively. Interpreting these results will give us an idea of the company's performance and potential growth in the next 5 years. If R(2) and R(4) have a positive value, it means that the company is expected to have a higher revenue compared to the previous year, while a negative value would mean a decrease in revenue. Question 5 Find the optimum value of this function using direct substitution method of constrained optimisation: Z = x ^ 2 + 0.5xy + y ^ 2 subject to the constraint 90 = y + 2x Answer The direct substitution method is a commonly used technique in economics for solving constrained optimization problems. In this method, we substitute the constraint equation into the objective function, reduce it to a single-variable function, and then solve for the optimum value. First, let's restate the problem in terms of the objective function and the constraint equation: Objective Function: Z = x^2 + 0.5xy + y^2 Constraint: 90 = y + 2x To use the direct substitution method, we need to manipulate the constraint equation to express one variable in terms of the other. In this case, we can express y in terms of x as: y = 90 - 2x. Now, we can substitute this expression for y into the objective function: Z = x^2 + 0.5x(90-2x) + (90-2x)^2 Expanding and simplifying: Z = x^2 + 45x - x^2 + 8100 - 180x + 4x^2 Z = 5x^2 - 135x + 8100 To find the optimum value of this function, we need to take the first derivative of Z with respect to x and set it equal to zero: dZ/dx = 10x - 135 = 0 x = 13.5 Next, we can plug this value of x back into the original constraint equation to solve for y: 90 = y + 2(13.5) y = 63 Therefore, the optimum value of the function Z is 13.5 and 63, where x = 13.5 and y = 63.