Question 1200016: Grace's bakery specializes in giant cupcakes. Bach cardboard cupcake container is filled with batter and baked in the oven. The baked cupcake forms a perfect hemispherical on top of the container. The surface area of this is frosted. A clear plastic lid, identical in size and shape to the cupcake container, is then placed on top. These containers are designed such that R, =
R, and h - 2R. Given that 100 cm° of batter costs 94, 100 cm° of frosting costs
4
274, 100 cm? of cardboard costs 1.5¢, and 100 cm? of plastic costs 34, complete the tasks below on the next page.
The following formulas may be helpful:
Conical frustum:
Hemisphere:
SA = 277?
2
V=
S4 (sides only) =兀（号十 )s
V-三Th(R+R及十吗)
Create functions of R, which return the cost, in dollars, of ingredients, I, and packaging, P.
Approximate all coefficients to 4 decimal places.
Finc the volume of a baked cupcake if Grace wants 85% of the cost to go toward ingredients. Round to 2 decimal places.
Hints: Consider that.
15
Use the "intersect" graphing calculator tool to find R,.
c. Find the efficiency ratio for the packaging to the baked cupcake, to 2 decimal places.
38
Found 2 solutions by textot, ikleyn:
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website!
**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.
Answer by ikleyn(52790)  (Show Source):
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