Subjects algebra

Subscription Prices

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1. **Problem Statement:** We need to find the price of a basic subscription and a premium subscription given the revenue function and contour lines on the graph. 2. **Understanding the Revenue Function:** The revenue $R$ is given by $$R = \text{price\_basic} \times c + \text{price\_premium} \times d,$$ where $c$ is the number of basic subscribers and $d$ is the number of premium subscribers. 3. **Analyzing the Graph:** The contour lines represent constant revenue values: 1000, 2000, 3000, 4000, and 5000. 4. **Interpreting Contour Spacing:** The increase from 1000 to 2000 corresponds to an increase of 1000 in revenue. 5. **Horizontal Movement (change in $c$):** Moving horizontally by 50 units in $c$ (with $d$ fixed) increases revenue by 1000. 6. **Vertical Movement (change in $d$):** Moving vertically by 100 units in $d$ (with $c$ fixed) also increases revenue by 1000. 7. **Calculating Price of Basic Subscription:** Since increasing $c$ by 50 increases revenue by 1000, $$\text{price\_basic} = \frac{1000}{50} = 20.$$ 8. **Calculating Price of Premium Subscription:** Since increasing $d$ by 100 increases revenue by 1000, $$\text{price\_premium} = \frac{1000}{100} = 10.$$ 9. **Final Answer:** - Price of basic subscription = 20 - Price of premium subscription = 10 This means each basic subscriber contributes 20 dollars to revenue, and each premium subscriber contributes 10 dollars.