Questions: Suppose that the relationship between price and quantity of gizmo kits we can buy is linear. When the price is 46, we can buy 1383 gizmo kits. If we lower the price we will pay to 10, we can buy only 504 kits. Find the equation of the line using the formula p-p0=m(q-q0). Where p0=46

Suppose that the relationship between price and quantity of gizmo kits we can buy is linear. When the price is 46, we can buy 1383 gizmo kits. If we lower the price we will pay to 10, we can buy only 504 kits. Find the equation of the line using the formula p-p0=m(q-q0). Where p0=46

Transcript text: Suppose that the relationship between price and quantity of gizmo kits we can buy is linear. When the price is $\$ 46$, we can buy 1383 gizmo kits. If we lower the price we will pay to $\$ 10$, we can buy only 504 kits. Find the equation of the line using the formula $p-p_{0}=m\left(q-q_{0}\right)$. Where $p_{0}=46$

Solution

Solution Steps

To find the equation of the line that represents the relationship between price (p) and quantity (q) of gizmo kits, we need to determine the slope (m) and use the point-slope form of the line equation. We are given two points: (q1, p1) = (1383, 46) and (q2, p2) = (504, 10). We can calculate the slope (m) using these points and then use the point-slope formula to find the equation.

Solution Approach

Calculate the slope (m) using the formula $ m = \frac{p2 - p1}{q2 - q1} $.
Use the point-slope form $ p - p0 = m(q - q0) $ to find the equation of the line.

Step 1: Calculate the Slope

Given two points $(q_1, p_1) = (1383, 46)$ and $(q_2, p_2) = (504, 10)$, we calculate the slope $m$ using the formula: \[ m = \frac{p_2 - p_1}{q_2 - q_1} \] Substituting the given values: \[ m = \frac{10 - 46}{504 - 1383} = \frac{-36}{-879} \approx 0.0410 \]

Step 2: Use the Point-Slope Form

Using the point-slope form of the line equation $p - p_0 = m(q - q_0)$ with $p_0 = 46$ and $q_0 = 1383$, we get: \[ p - 46 = 0.0410 (q - 1383) \]

Step 3: Simplify the Equation

Simplify the equation to find $p$ in terms of $q$: \[ p = 46 + 0.0410 (q - 1383) \] \[ p = 46 + 0.0410q - 56.703 \] \[ p = 0.0410q - 10.703 \]

Step 4: Calculate the Price for a Given Quantity

To find the price when the quantity is 1000, substitute $q = 1000$ into the equation: \[ p = 0.0410 \times 1000 - 10.703 \] \[ p = 41 - 10.703 \] \[ p \approx 30.31 \]