Question 5: It is given that x and y are in direct proportion and the difference in y, when x is 5 and 13, is 48. (a) Find the equation connecting x and y. (b) Find the value of x... Question 5: It is given that x and y are in direct proportion and the difference in y, when x is 5 and 13, is 48. (a) Find the equation connecting x and y. (b) Find the value of x when y = 72. Question 12: The volume, V cm³, of an object is directly proportional to the cube of its diameter, d cm. Given that V = 32.5 when d = 5, (a) find a formula connecting V and d, (b) calculate the diameter of the object when it has a volume of 89 cm³, (c) write down a possible shape of the object, giving a reason for your answer.
Understand the Problem
The questions revolve around direct proportion and involve finding equations and calculations based on provided values. Specifically, they require establishing relationships between variables and performing calculations based on those relationships.
Answer
The equation is $y = 6x$; when $y = 72$, then $x = 12$.
Answer for screen readers
The equation connecting $x$ and $y$ is $y = 6x$, and when $y = 72$, $x = 12$.
Steps to Solve
- Understand Direct Proportion
Since $x$ and $y$ are in direct proportion, we can express this relationship as: $$ y = kx $$ where $k$ is the constant of proportionality.
- Find the Constant of Proportionality
We need to find the constant $k$. We're given that the difference in $y$ when $x$ changes from 5 to 13 is 48.
Calculate $y$ for both values:
- When $x = 5$: $$ y_1 = k \cdot 5 $$
- When $x = 13$: $$ y_2 = k \cdot 13 $$
The difference is given by: $$ y_2 - y_1 = 48 $$
Substituting in the equations: $$ (k \cdot 13) - (k \cdot 5) = 48 $$
Simplifying: $$ 8k = 48 $$
- Solve for k
Dividing both sides by 8: $$ k = \frac{48}{8} = 6 $$
- Write the Equation Connecting x and y
Now that we have $k$, we can write the equation: $$ y = 6x $$
- Find x when y = 72
Use the equation $y = 6x$ to find $x$ for $y = 72$: $$ 72 = 6x $$
Now, solve for $x$: $$ x = \frac{72}{6} = 12 $$
The equation connecting $x$ and $y$ is $y = 6x$, and when $y = 72$, $x = 12$.
More Information
Understanding direct proportion is essential in various fields, including physics and economics, where relationships between variables often depend on a constant ratio.
Tips
- Misunderstanding Proportionality: Sometimes, users confuse direct proportionality with inverse or other relations. Always remember $y = kx$ for direct proportion.
- Algebra Mistakes: Errors in simplifying equations can occur. Ensure careful arithmetic operations.
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