document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200016>Question 1200016</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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, =<BR>\n" );
document.write( "R, and h - 2R. Given that 100 cm° of batter costs 94, 100 cm° of frosting costs<BR>\n" );
document.write( "4<BR>\n" );
document.write( "274, 100 cm? of cardboard costs 1.5¢, and 100 cm? of plastic costs 34, complete the tasks below on the next page.<BR>\n" );
document.write( "The following formulas may be helpful:<BR>\n" );
document.write( "Conical frustum:<BR>\n" );
document.write( "Hemisphere:<BR>\n" );
document.write( "SA = 277?<BR>\n" );
document.write( "2<BR>\n" );
document.write( "V=<BR>\n" );
document.write( "S4 (sides only) =兀（号十 )s<BR>\n" );
document.write( "V-三Th(R+R及十吗)<BR>\n" );
document.write( "Create functions of R, which return the cost, in dollars, of ingredients, I, and packaging, P.<BR>\n" );
document.write( "Approximate all coefficients to 4 decimal places.<BR>\n" );
document.write( "Finc the volume of a baked cupcake if Grace wants 85% of the cost to go toward ingredients. Round to 2 decimal places.<BR>\n" );
document.write( "Hints: Consider that.<BR>\n" );
document.write( "15<BR>\n" );
document.write( "Use the \"intersect\" graphing calculator tool to find R,.<BR>\n" );
document.write( "c. Find the efficiency ratio for the packaging to the baked cupcake, to 2 decimal places.<BR>\n" );
document.write( "38 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848133 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848133\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "**1. Define Functions**\r<BR>\n" );
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document.write( "* **Cost of Ingredients (I)**:\r<BR>\n" );
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document.write( "   - Cost of batter: <BR>\n" );
document.write( "      - Volume of batter = Volume of hemisphere = (2/3) * π * R^3<BR>\n" );
document.write( "      - Cost per 100 cm³ = $0.09 <BR>\n" );
document.write( "      - Cost of batter = (2/3) * π * R^3 * 0.09 / 100 <BR>\n" );
document.write( "   - Cost of frosting: <BR>\n" );
document.write( "      - Surface area of hemisphere = 2 * π * R^2<BR>\n" );
document.write( "      - Cost per 100 cm² = $0.27<BR>\n" );
document.write( "      - Cost of frosting = 2 * π * R^2 * 0.27 / 100\r<BR>\n" );
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document.write( "   - **I(R) = (0.0001885 * π * R^3) + (0.0054 * π * R^2)**\r<BR>\n" );
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document.write( "* **Cost of Packaging (P)**:\r<BR>\n" );
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document.write( "   - Cost of cardboard: <BR>\n" );
document.write( "      - Surface area of cardboard = 2 * π * R * h + π * R^2 <BR>\n" );
document.write( "         - Substitute h = 2R: <BR>\n" );
document.write( "            - Surface area = 2 * π * R * (2R) + π * R^2 = 5 * π * R^2<BR>\n" );
document.write( "      - Cost per 100 cm² = $0.015<BR>\n" );
document.write( "      - Cost of cardboard = 5 * π * R^2 * 0.015 / 100 \r<BR>\n" );
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document.write( "   - Cost of plastic lid: <BR>\n" );
document.write( "      - Surface area of plastic lid = π * R^2<BR>\n" );
document.write( "      - Cost per 100 cm² = $0.03<BR>\n" );
document.write( "      - Cost of plastic lid = π * R^2 * 0.03 / 100\r<BR>\n" );
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document.write( "   - **P(R) = (0.00075 * π * R^2) + (0.0003 * π * R^2) = 0.00105 * π * R^2**\r<BR>\n" );
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document.write( "**2. Find Volume for 85% Ingredient Cost**\r<BR>\n" );
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document.write( "* **Set up the equation:** <BR>\n" );
document.write( "   - I(R) / (I(R) + P(R)) = 0.85\r<BR>\n" );
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document.write( "* **Substitute I(R) and P(R):**<BR>\n" );
document.write( "   - [(0.0001885 * π * R^3) + (0.0054 * π * R^2)] / <BR>\n" );
document.write( "     [(0.0001885 * π * R^3) + (0.0054 * π * R^2) + (0.00105 * π * R^2)] = 0.85\r<BR>\n" );
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document.write( "* **Use a graphing calculator to find the intersection point:** <BR>\n" );
document.write( "   - Graph the left-hand side and the right-hand side of the equation. <BR>\n" );
document.write( "   - Find the x-coordinate (R) of the intersection point.\r<BR>\n" );
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document.write( "* **Calculate the volume of the cupcake:** <BR>\n" );
document.write( "   - Volume = (2/3) * π * R^3 \r<BR>\n" );
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document.write( "**3. Find Efficiency Ratio**\r<BR>\n" );
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document.write( "* **Efficiency Ratio = Volume of Cupcake / Volume of Packaging**\r<BR>\n" );
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document.write( "   - Volume of Packaging: <BR>\n" );
document.write( "      - Consider the volume of the cardboard container as the packaging volume.<BR>\n" );
document.write( "      - Volume of container = π * R^2 * h = π * R^2 * (2R) = 2 * π * R^3\r<BR>\n" );
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document.write( "   - Efficiency Ratio = [(2/3) * π * R^3] / [2 * π * R^3] = 1/3\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **Efficiency Ratio = 0.33**\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* 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.<BR>\n" );
document.write( "* The efficiency ratio indicates that for every 1 unit of packaging volume, there are 0.33 units of cupcake volume.<BR>\n" );
document.write( "**1. Define Functions for Cost**\r<BR>\n" );
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document.write( "* **Cost of Ingredients (I)**\r<BR>\n" );
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document.write( "   * Cost of batter: <BR>\n" );
document.write( "      * Volume of batter = Volume of hemisphere = (2/3) * π * R^3<BR>\n" );
document.write( "      * Cost of batter = (2/3) * π * R^3 * (9 cents / 100 cm^3) <BR>\n" );
document.write( "      * Cost of batter = 0.0188 * R^3 \r<BR>\n" );
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document.write( "   * Cost of frosting: <BR>\n" );
document.write( "      * Surface area of hemisphere = 2 * π * R^2<BR>\n" );
document.write( "      * Cost of frosting = 2 * π * R^2 * (27 cents / 100 cm^2) <BR>\n" );
document.write( "      * Cost of frosting = 0.1696 * R^2\r<BR>\n" );
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document.write( "   * **I(R) = 0.0188 * R^3 + 0.1696 * R^2** \r<BR>\n" );
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document.write( "* **Cost of Packaging (P)**\r<BR>\n" );
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document.write( "   * Cost of cardboard: <BR>\n" );
document.write( "      * Surface area of cardboard = 2 * π * R^2 <BR>\n" );
document.write( "      * Cost of cardboard = 2 * π * R^2 * (1.5 cents / 100 cm^2)<BR>\n" );
document.write( "      * Cost of cardboard = 0.0094 * R^2\r<BR>\n" );
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document.write( "   * Cost of plastic lid: <BR>\n" );
document.write( "      * Surface area of plastic = 2 * π * R^2 <BR>\n" );
document.write( "      * Cost of plastic = 2 * π * R^2 * (3 cents / 100 cm^2)<BR>\n" );
document.write( "      * Cost of plastic = 0.0189 * R^2\r<BR>\n" );
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document.write( "   * **P(R) = 0.0094 * R^2 + 0.0189 * R^2 = 0.0283 * R^2**\r<BR>\n" );
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document.write( "**2. Find the Volume for 85% Ingredient Cost**\r<BR>\n" );
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document.write( "* **Set up the equation:** <BR>\n" );
document.write( "   * I(R) / (I(R) + P(R)) = 0.85\r<BR>\n" );
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document.write( "* **Substitute the functions:**<BR>\n" );
document.write( "   * (0.0188 * R^3 + 0.1696 * R^2) / (0.0188 * R^3 + 0.1696 * R^2 + 0.0283 * R^2) = 0.85\r<BR>\n" );
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document.write( "* **Simplify and solve for R:**<BR>\n" );
document.write( "   * 0.0188 * R^3 + 0.1696 * R^2 = 0.85 * (0.0188 * R^3 + 0.1979 * R^2)<BR>\n" );
document.write( "   * 0.00254 * R^3 = 0.0265 * R^2<BR>\n" );
document.write( "   * R = 0.0265 / 0.00254 <BR>\n" );
document.write( "   * R ≈ 10.43 cm\r<BR>\n" );
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document.write( "* **Calculate the volume of the cupcake:**<BR>\n" );
document.write( "   * Volume = (2/3) * π * R^3 = (2/3) * π * (10.43 cm)^3 <BR>\n" );
document.write( "   * Volume ≈ 2380.94 cm^3\r<BR>\n" );
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document.write( "**3. Find the Efficiency Ratio**\r<BR>\n" );
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document.write( "* **Efficiency Ratio = Volume of Cupcake / Volume of Packaging**\r<BR>\n" );
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document.write( "   * Volume of Packaging = Volume of Hemisphere + Volume of Cylinder (height = R)<BR>\n" );
document.write( "      * Volume of Cylinder = π * R^2 * h = π * R^2 * 2R = 2 * π * R^3<BR>\n" );
document.write( "      * Volume of Packaging = (2/3) * π * R^3 + 2 * π * R^3 = (8/3) * π * R^3\r<BR>\n" );
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document.write( "   * Efficiency Ratio = [(2/3) * π * R^3] / [(8/3) * π * R^3] = 1/4 <BR>\n" );
document.write( "   * Efficiency Ratio = 0.25\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **I(R) = 0.0188 * R^3 + 0.1696 * R^2**<BR>\n" );
document.write( "* **P(R) = 0.0283 * R^2**<BR>\n" );
document.write( "* **Volume of Cupcake for 85% Ingredient Cost ≈ 2380.94 cm^3**<BR>\n" );
document.write( "* **Efficiency Ratio = 0.25** \r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This solution assumes the cupcake container is a perfect cylinder. <BR>\n" );
document.write( "* The \"intersect\" graphing calculator tool can be used to find the value of R that satisfies the equation in step 2 more accurately.<BR>\n" );
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