## Explanation: Least Average Earnings from Films with Female Leads Let's look at the table you provided, specifically the column for films with female leads: | Distributor | Global Rev ($M) | # films | $/film ($M) | |--------------------------|-----------------|---------|-------------| | Walt Disney Studios | 4099.5 | 4 | 102. | | Universal Pictures | 892. | 9 | 99.2 | | 20th Century Fox | 764.5 | 4 | 191.1 | | Sony Pictures | 577.4 | 5 | 115.8 | | Lionsgate | 581.4 | 5 | 116.2 | | Paramount Pictures | 481.3 | 5 | 101.8 | | Warner Bros. Pictures | 402.4 | 3 | 133.7 | ### Step-by-Step: 1. **Find the column "$/film ($M)"** This column shows the average revenue per film with a female lead for each studio. 2. **Compare the values:** - Walt Disney Studios: 102. - Universal Pictures: 99.2 - 20th Century Fox: 191.1 - Sony Pictures: 115.8 - Lionsgate: 116.2 - Paramount Pictures: 101.8 - Warner Bros. Pictures: 133.7 3. **Identify the lowest number:** - The lowest average is **99.2** (Universal Pictures). ### **Answer** - **Universal Pictures** earned the least on average from films with female leads (**$99.2 million per film**). If you have any more questions about how to read the table, just ask!

## Present Value Calculation To find the present value (PV) of \$50,000 received in 5 years with a discount rate of 8% compounded annually, we use the formula: \[ PV = \frac{FV}{(1 + r)^n} \] Where: - \( FV \) = Future Value = \$50,000 - \( r \) = discount rate = 0.08 - \( n \) = number of years = 5 ### Step-by-Step Calculation 1. **Calculate \((1 + r)^n\)**: \[ (1 + 0.08)^5 = (1.08)^5 \approx 1.4693 \] 2. **Substitute into the PV formula**: \[ PV = \frac{50000}{1.4693} \approx 34027.19 \] 3. **Round to the nearest whole number**: \[ PV \approx 34027 \] ### **Final Answer** - The present value today is **$34,027**. ### **Options:** - A. $34,027 - B. $34,030 - C. $34,045 - D. $34,100 **Correct Option: A. $34,027**

## Defects per Million Opportunities (DPMO) Calculation To calculate DPMO, we use the formula: \[ DPMO = \left( \frac{Number \ of \ Defects}{Total \ Opportunities} \right) \times 1,000,000 \] ### Step-by-Step Calculation 1. **Identify the parameters**: - Number of units produced = 1,000 - Opportunities for defects per unit = 5 - Number of defects found = 25 2. **Calculate total opportunities**: \[ Total \ Opportunities = Number \ of \ Units \times Opportunities \ per \ Unit = 1000 \times 5 = 5000 \] 3. **Substitute into the DPMO formula**: \[ DPMO = \left( \frac{25}{5000} \right) \times 1,000,000 \] 4. **Calculate DPMO**: \[ DPMO = \left( 0.005 \right) \times 1,000,000 = 5000 \] ### **Final Answer** - The DPMO is **5,000**. If you need further clarification or assistance, feel free to ask!

## Ionic Strength and Activity Calculation To find the ionic strength of a solution and the activities of ions, we can follow these steps: ### Step 1: Calculate Ionic Strength The formula for ionic strength (\(I\)) is: \[ I = \frac{1}{2} \sum c_i z_i^2 \] Where: - \(c_i\) = concentration of each ion (mol/L) - \(z_i\) = charge of each ion ### Given Information - Concentration of MgCl₂ = 8.0 mM = 0.008 M - Dissociation of MgCl₂ in solution: - \(MgCl_2 \rightarrow Mg^{2+} + 2Cl^{-}\) ### Step 2: Determine Ion Concentrations - Concentration of \(Mg^{2+}\) = 0.008 M - Concentration of \(Cl^{-}\) = \(2 \times 0.008 = 0.016\) M ### Step 3: Calculate Ionic Strength Now substitute into the ionic strength formula: \[ I = \frac{1}{2} \left( c_{Mg^{2+}} z_{Mg^{2+}}^2 + c_{Cl^{-}} z_{Cl^{-}}^2 \right) \] Where: - \(z_{Mg^{2+}} = +2\) - \(z_{Cl^{-}} = -1\) Substituting the values: \[ I = \frac{1}{2} \left( 0.008 \times (2)^2 + 0.016 \times (1)^2 \right) \] Calculating: \[ I = \frac{1}{2} \left( 0.008 \times 4 + 0.016 \times 1 \right) \] \[ = \frac{1}{2} \left( 0.032 + 0.016 \right) = \frac{1}{2} \times 0.048 = 0.024 \text{ M} \] ### Step 4: Calculate Activities The activity (\(a\)) of an ion is related to its concentration (\(c\)) and activity coefficient (\(\gamma\)): \[ a = \gamma \cdot c \] Activity coefficients can be estimated using models such as the Debye-Hückel equation, but for simplicity, we will assume \(\gamma\) is close to 1 for dilute solutions. #### For \(Mg^{2+}\): \[ a_{Mg^{2+}} \approx 1 \cdot 0.008 = 0.008 \] #### For \(Cl^{-}\): \[ a_{Cl^{-}} \approx 1 \cdot 0.016 = 0.016 \] ### **Final Answers** - **Ionic Strength (I)**: **0.024 M** - **Activity of \(Mg^{2+}\)**: **0.008** - **Activity of \(Cl^{-}\)**: **0.016** If you need further assistance, feel free to ask!