Mathematics Answer Note #7

Question 1 Students measured objects and displayed their data on the line below if you put all of the objects together and and what would be the total length of the objects? Answer The total length of the objects can be calculated by adding together the lengths of each individual object. First, we must identify the units of measurement used for each object to ensure accurate calculations. Once we have the units, we can proceed with finding the total length. By utilizing these resources and applying the concepts learned, we can accurately calculate the total length of the objects measured by the students. It is important for students to have a thorough understanding of measurement and data representation in order to effectively analyze and interpret information presented in various forms. Question 2 10. If both cities continue to give raises as described above, will Norwalk or Madison reach an hourly minimum wage of at least (l) $29.50 first? You may use algebra or trialand-error to determine your solution, but you must show your problem-solving process in detail. Write your conclusion in complete sentences. (10 pt) 29.50=15.00×(1+0.05)t 29.50/15.00=(1+0.05)t29.50/15.00=(1+0.05)t 1.9667=(1.05)t1.9667=(1.05)t ln(1.9667)=t × ln(1.05) t=ln⁡(1.9667)ln(1.05)t=ln(1.05)ln(1.9667) t=0.68/0.0488 t=14.11 so it will take 14 years for Norwalk to reach an hourly minimum wage of at least $29.50 Answer To determine which city will reach an hourly minimum wage of at least $29.50 first, we can use algebra to solve the equations. Let's first review the information given in the problem. Both cities, Norwalk and Madison, are currently at an hourly minimum wage of $15.00, and they are both giving a 5% raise each year. This can be represented by the equations: Norwalk: $15.00(1+0.05)^t Madison: $15.00(1+0.05)^t We can simplify these equations to: Norwalk: $15.75^t Madison: $15.75^t Next, we want to find the value of t, or the number of years, when each city's minimum wage reaches $29.50. We can set up an equation as follows: $29.50 = $15.75^t To solve this equation, we can take the natural log (ln) of both sides and use the logarithmic property of ln(a^b) = bln(a). This will allow us to isolate the value of t. ln(29.50) = t ln(15.75) We can simplify the equation to: t = ln(29.50)/ln(15.75) Using a calculator, we find that t is approximately equal to 14.11. This means that it will take approximately 14.11 years for both Norwalk and Madison to reach an hourly minimum wage of at least $29.50. However, the question asks which city will reach this wage first. To find this, we must determine which city will reach this wage in fewer years. Since both cities are increasing their minimum wage by the same percentage each year, the city with the higher starting wage will reach $29.50 first. In this case, it will be Norwalk, as its starting wage is $15.00, compared to Madison's starting wage of $15.00. Therefore, it can be concluded that Norwalk will reach an hourly minimum wage of at least $29.50 first, in approximately 14 years. Question 3 Harsha is 40yrs old and ritu is 60yrs old. How many year ago was the ratio of their ages 3:5 ? Answer The ratio of two ages is the proportion of one age to the other. So, to find the number of years ago the ratio of their ages was 3:5, we need to determine the difference between their current ages and their ages at the time when the ratio was 3:5. To calculate this, we can follow the steps below: Step 1: Find the current ages of Harsha and Ritu. Given that Harsha is currently 40 years old and Ritu is currently 60 years old, their current ages can be represented as H = 40 and R = 60. Step 2: Set up a ratio proportion using their current ages. The ratio of their ages is represented as 3:5. So, we can set up the following proportion: H/R = 3/5 Step 3: Solve for their ages when the ratio was 3:5. Cross-multiply the ratio proportion from Step 2 to get: 5H = 3R Step 4: Substitute the current ages from Step 1 into the equation in Step 3. 5(40) = 3(60) Step 5: Simplify the equation. 200 = 180 Step 6: Subtract the current ages from their ages when the ratio was 3:5. 60 - 40 = 20 This shows that 20 years ago, the ratio of Harsha and Ritu's ages was 3:5. In addition, we can also solve this problem using the concept of proportionality. Since the ratio of Harsha and Ritu's ages is 3:5, we can represent their ages as 3x and 5x. Therefore, their current ages can be written as: Harsha's age = 3x = 40 Ritu's age = 5x = 60 Solving for x: 3x = 40 x = 40/3 = 13.33 Therefore, 20 years ago, Harsha's age was 3(13.33) = 40 and Ritu's age was 5(13.33) = 66.65, giving us a ratio of approximately 3:5. Question 4 If \( \mathrm{f}(-8)=-6 \), write a corresponding ordered-pair solution. What is the corresponding ordered-pair? \( \square \) (Type an ordered pair.) Answer According to the given information, the function \( \mathrm{f} \) has an input of -8 and an output of -6. This can be written as an ordered pair \( (-8, -6) \), where the first value represents the input and the second value represents the output. To verify this solution, we can plug in -8 as the input for the given function and see if we get an output of -6. So, \( \mathrm{f}(-8) = -6 \) is indeed satisfied and the corresponding ordered-pair solution is \( (-8, -6) \). Question 5 When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out? $\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$ Answer After carefully reviewing the provided question, the answer to this question is option $textbf{(A) } 5$. This can be determined through a few mathematical calculations and logical reasoning. Firstly, let us calculate the sum of all the integers from $1$ to $9$. This can be done by using the formula for finding the sum of an arithmetic sequence, which is $S_n = n/2(a_1 + a_n)$, where $n$ is the number of terms and $a_1$ and $a_n$ are the first and last terms respectively. In this case, $n = 9$, $a_1 = 1$ and $a_n = 9$. Substituting these values in the formula, we get $S_n = 9/2(1+9) = 45$. Therefore, the sum of all the integers from $1$ to $9$ is $45$. Now, let us assume that the number that Yunji left out is $x$. Therefore, the incorrect sum that she calculated would be $45-x$. As per the question, this incorrect sum turned out to be a square number. This means that there exists a number $n$ such that $n^2 = 45-x$. By rearranging this equation, we get $x = 45-n^2$. In a perfect scenario where all the integers from $1$ to $9$ are included, the correct sum would be a perfect square number. This can also be determined by examining the given options. Out of the given options, only $6$ and $8$ are perfect squares. This means that the correct sum must either be $6$ or $8$. However, $6$ is not a possible option as we have already calculated the sum to be $45$. Hence, the only possibility left is $8$. Therefore, $45-n^2 = 8 implies n^2 = 45-8 = 37$. However, $37$ is not a perfect square as the possible perfect squares between $30$ and $40$ are $36$ and $49$. Thus, we can conclude that $n^2 = 37$ is not possible. Furthermore, after examining the given options, we can see that $37$ is not a possible answer as $37$ is not included in the given options. This leaves us with only one possibility which is $n^2 = 49$. Therefore, the number that Yunji left out is $x = 45-49 = textbf{5}$.