Question 1200016


**1. Define Functions**

* **Cost of Ingredients (I)**:

   - Cost of batter: 
      - Volume of batter = Volume of hemisphere = (2/3) * π * R^3
      - Cost per 100 cm³ = $0.09 
      - Cost of batter = (2/3) * π * R^3 * 0.09 / 100 
   - Cost of frosting: 
      - Surface area of hemisphere = 2 * π * R^2
      - Cost per 100 cm² = $0.27
      - Cost of frosting = 2 * π * R^2 * 0.27 / 100

   - **I(R) = (0.0001885 * π * R^3) + (0.0054 * π * R^2)**

* **Cost of Packaging (P)**:

   - Cost of cardboard: 
      - Surface area of cardboard = 2 * π * R * h + π * R^2 
         - Substitute h = 2R: 
            - Surface area = 2 * π * R * (2R) + π * R^2 = 5 * π * R^2
      - Cost per 100 cm² = $0.015
      - Cost of cardboard = 5 * π * R^2 * 0.015 / 100 

   - Cost of plastic lid: 
      - Surface area of plastic lid = π * R^2
      - Cost per 100 cm² = $0.03
      - Cost of plastic lid = π * R^2 * 0.03 / 100

   - **P(R) = (0.00075 * π * R^2) + (0.0003 * π * R^2) = 0.00105 * π * R^2**

**2. Find Volume for 85% Ingredient Cost**

* **Set up the equation:** 
   - I(R) / (I(R) + P(R)) = 0.85

* **Substitute I(R) and P(R):**
   - [(0.0001885 * π * R^3) + (0.0054 * π * R^2)] / 
     [(0.0001885 * π * R^3) + (0.0054 * π * R^2) + (0.00105 * π * R^2)] = 0.85

* **Use a graphing calculator to find the intersection point:** 
   - Graph the left-hand side and the right-hand side of the equation. 
   - Find the x-coordinate (R) of the intersection point.

* **Calculate the volume of the cupcake:** 
   - Volume = (2/3) * π * R^3 

**3. Find Efficiency Ratio**

* **Efficiency Ratio = Volume of Cupcake / Volume of Packaging**

   - Volume of Packaging: 
      - Consider the volume of the cardboard container as the packaging volume.
      - Volume of container = π * R^2 * h = π * R^2 * (2R) = 2 * π * R^3

   - Efficiency Ratio = [(2/3) * π * R^3] / [2 * π * R^3] = 1/3

**Therefore:**

* **Efficiency Ratio = 0.33**

**Note:**

* This solution provides the framework and steps. You'll need to use a graphing calculator to find the specific values for R and the volume of the cupcake.
* The efficiency ratio indicates that for every 1 unit of packaging volume, there are 0.33 units of cupcake volume.
**1. Define Functions for Cost**

* **Cost of Ingredients (I)**

   * Cost of batter: 
      * Volume of batter = Volume of hemisphere = (2/3) * π * R^3
      * Cost of batter = (2/3) * π * R^3 * (9 cents / 100 cm^3) 
      * Cost of batter = 0.0188 * R^3 

   * Cost of frosting: 
      * Surface area of hemisphere = 2 * π * R^2
      * Cost of frosting = 2 * π * R^2 * (27 cents / 100 cm^2) 
      * Cost of frosting = 0.1696 * R^2

   * **I(R) = 0.0188 * R^3 + 0.1696 * R^2** 

* **Cost of Packaging (P)**

   * Cost of cardboard: 
      * Surface area of cardboard = 2 * π * R^2 
      * Cost of cardboard = 2 * π * R^2 * (1.5 cents / 100 cm^2)
      * Cost of cardboard = 0.0094 * R^2

   * Cost of plastic lid: 
      * Surface area of plastic = 2 * π * R^2 
      * Cost of plastic = 2 * π * R^2 * (3 cents / 100 cm^2)
      * Cost of plastic = 0.0189 * R^2

   * **P(R) = 0.0094 * R^2 + 0.0189 * R^2 = 0.0283 * R^2**

**2. Find the Volume for 85% Ingredient Cost**

* **Set up the equation:** 
   * I(R) / (I(R) + P(R)) = 0.85

* **Substitute the functions:**
   * (0.0188 * R^3 + 0.1696 * R^2) / (0.0188 * R^3 + 0.1696 * R^2 + 0.0283 * R^2) = 0.85

* **Simplify and solve for R:**
   * 0.0188 * R^3 + 0.1696 * R^2 = 0.85 * (0.0188 * R^3 + 0.1979 * R^2)
   * 0.00254 * R^3 = 0.0265 * R^2
   * R = 0.0265 / 0.00254 
   * R ≈ 10.43 cm

* **Calculate the volume of the cupcake:**
   * Volume = (2/3) * π * R^3 = (2/3) * π * (10.43 cm)^3 
   * Volume ≈ 2380.94 cm^3

**3. Find the Efficiency Ratio**

* **Efficiency Ratio = Volume of Cupcake / Volume of Packaging**

   * Volume of Packaging = Volume of Hemisphere + Volume of Cylinder (height = R)
      * Volume of Cylinder = π * R^2 * h = π * R^2 * 2R = 2 * π * R^3
      * Volume of Packaging = (2/3) * π * R^3 + 2 * π * R^3 = (8/3) * π * R^3

   * Efficiency Ratio = [(2/3) * π * R^3] / [(8/3) * π * R^3] = 1/4 
   * Efficiency Ratio = 0.25

**Therefore:**

* **I(R) = 0.0188 * R^3 + 0.1696 * R^2**
* **P(R) = 0.0283 * R^2**
* **Volume of Cupcake for 85% Ingredient Cost ≈ 2380.94 cm^3**
* **Efficiency Ratio = 0.25** 

**Note:**

* This solution assumes the cupcake container is a perfect cylinder. 
* The "intersect" graphing calculator tool can be used to find the value of R that satisfies the equation in step 2 more accurately.