Questions: 36. x is directly proportional to the cube of y. x=3 when y=7. What is x when y=14? x= 1. 24 2. 196 3. 12 4. 21 37. 8x+27+3y=27y+3; x= 1. 13y+3 2. 3y-3 3. 3y 4. 6

Transcript text: 36. x is directly proportional to the cube of y . $\mathrm{x}=3$ when $\mathrm{y}=7$. What is x when $\mathrm{y}=14$ ? $\mathrm{x}=$ 1 24 2 196 3 12 4 21 37. $8 x+27+3 y=27 y+3 ; \quad x=$ 1 $13 y+3$ 2 $3 y-3$ 3 $3 y$ 4 6

Solution

Solution Steps

Solution Approach

For the equation $8 + 7 = 5x - 15$, solve for $x$ by isolating it on one side of the equation.
For the problem where $x$ is directly proportional to the cube of $y$, use the proportionality relationship to find the constant of proportionality and then calculate $x$ for $y = 14$.
For the equation $8x + 27 + 3y = 27y + 3$, rearrange the terms to solve for $x$ in terms of $y$.

Step 1: Solve the Equation $8 + 7 = 5x - 15$

Starting with the equation: \[ 8 + 7 = 5x - 15 \] we simplify it to: \[ 15 = 5x - 15 \] Adding 15 to both sides gives: \[ 30 = 5x \] Dividing both sides by 5 results in: \[ x = 6 \]

Step 2: Find $x$ When $y = 14$ Given $x$ is Proportional to $y^3$

Given that $x$ is directly proportional to the cube of $y$, we can express this as: \[ x = k y^3 \] where $k$ is the constant of proportionality. We know that $x = 3$ when $y = 7$: \[ 3 = k (7^3) \implies k = \frac{3}{343} \approx 0.008746355685131196 \] Now, substituting $y = 14$: \[ x = k (14^3) = 0.008746355685131196 \times 2744 = 24.0 \]

Step 3: Solve the Equation $8x + 27 + 3y = 27y + 3$

Starting with the equation: \[ 8x + 27 + 3y = 27y + 3 \] Rearranging gives: \[ 8x = 27y - 3y + 3 - 27 \] which simplifies to: \[ 8x = 24y - 24 \] Dividing both sides by 8 results in: \[ x = 3y - 3 \]