Math Problem Statement
Solve
Two variables x and y are related such that y varies inversely as the square of x. (a) Write an equation in x, y, and k to describe the inverse variation, where k is the constant of variation. (b) Using the information in the table, calculate the value of (i) k, the constant of variation (ii) r (iii) f.
Also, the table below shows corresponding values of variables x and y, where y varies directly as x. Calculate the values of a and b.
Solution
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Question 1:
Part (a):
The problem states that "y varies inversely as the square of ." This means:
where is the constant of variation.
Part (b):
We are given a table of values for and :
\hline x & 3 & 1.8 & f \\ \hline y & 2 & r & 8 \\ \hline \end{array}$$ We need to calculate: 1. $$k$$, the constant of variation. 2. $$r$$. 3. $$f$$. We can use the equation $$y = \frac{k}{x^2}$$ and the given values to find these. --- **Step 1: Calculate $$k$$ using the first pair $$x = 3$$ and $$y = 2$$:** $$y = \frac{k}{x^2}$$ $$2 = \frac{k}{3^2} \quad \Rightarrow \quad 2 = \frac{k}{9} \quad \Rightarrow \quad k = 18$$ So, $$k = 18$$. --- **Step 2: Calculate $$r$$ when $$x = 1.8$$:** Using $$y = \frac{k}{x^2}$$, with $$k = 18$$ and $$x = 1.8$$, $$r = \frac{18}{1.8^2} = \frac{18}{3.24} \approx 5.56$$ --- **Step 3: Calculate $$f$$ when $$y = 8$$:** Using $$y = \frac{k}{x^2}$$, with $$y = 8$$ and $$k = 18$$, $$8 = \frac{18}{f^2} \quad \Rightarrow \quad f^2 = \frac{18}{8} = 2.25 \quad \Rightarrow \quad f = \sqrt{2.25} = 1.5$$ So, $$f = 1.5$$. --- ### Question 2: The problem says $$y$$ varies directly as $$x$$, which means: $$y = kx$$ We are given the following values: $$\begin{array}{|c|c|c|c|} \hline x & 2 & 5 & b \\ \hline y & 12 & a & 48 \\ \hline \end{array}$$ #### Step 1: Find the constant $$k$$ using $$x = 2$$ and $$y = 12$$: $$y = kx \quad \Rightarrow \quad 12 = 2k \quad \Rightarrow \quad k = 6$$ #### Step 2: Calculate $$a$$ when $$x = 5$$: $$a = 6 \times 5 = 30$$ #### Step 3: Calculate $$b$$ when $$y = 48$$: $$48 = 6 \times b \quad \Rightarrow \quad b = \frac{48}{6} = 8$$ --- ### Final Answers: 1. **For Question 1:** - $$k = 18$$ - $$r \approx 5.56$$ - $$f = 1.5$$ 2. **For Question 2:** - $$a = 30$$ - $$b = 8$$ Would you like more details on any of these steps? --- Here are 5 related questions you might consider: 1. How would the equation change if $$y$$ varied directly with the square of $$x$$? 2. Can you solve for $$x$$ in terms of $$y$$ for an inverse square variation? 3. What happens to $$y$$ if $$x$$ is halved in the direct variation equation? 4. How would you interpret an inverse cube relationship between $$y$$ and $$x$$? 5. How can we determine the units of $$k$$ based on real-world problems? **Tip:** In direct variation problems, solving for the constant $$k$$ first simplifies solving for unknown values of $$x$$ or $$y$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Inverse Variation
Direct Variation
Formulas
y = k / x^2 for inverse variation
y = kx for direct variation
Theorems
Inverse Variation Theorem
Direct Variation Theorem
Suitable Grade Level
Grade 10