Surface Area - Compound Shapes & Compound Interest

Surface Area - Compound Shapes

Problems and Figures

  • Round each answer to two decimal places. Use π = 3.14.
      - Dimensions for each shape identified:
        - Shape 1: Height = 5 in, Width = 7 in
        - Shape 2: Height = 10 in, Width = 6 cm
        - Shape 3: Height = 8 ft, Width = 6 cm
        - Shape 4: Height = 11 in, Width = 14 in
        - Shape 5: Height = 17 in, Width = 11 in
        - Shape 6: Height = 10 cm, Width = 19 ft
        - Shape 7: Height = 12 ft, Width = 16 ft
        - Shape 8: Height = 5 ft, Width = 6 mm
        - Shape 9: Height = 9 m, Width = 15 mm
        - Shape 10: Height = 3 m, Width = 6 mm
        - Shape 11: Height = 9 mm, Width = 5 mm
        - Shape 12: Height = 5 cm, Width = 12 cm

Surface Area Calculation

  • Surface area calculations must be performed for each shape identified above. Each formula relevant for rounding must include
      - Example Calculation:
        - For a rectangular prism:
          extSurfaceArea=2(lw+lh+wh)ext{Surface Area} = 2(lw + lh + wh)
          - Where l = length, w = width, h = height.

Compound Interest

Worksheets

  • Student Name: Colton Mcfeeters

Investment and Loan Calculations

  1. Investment Calculation:
       - Initial Investment: $52,400
       - Interest Rate: 6%
       - Compounding Period: Annually
       - Time: 5 years
       - Calculation:
         - Using formula:
         extTotalAmount=P(1+r)text{Total Amount} = P(1 + r)^t
         - Where P = principal, r = interest rate, t = time in years.
         - Substitute:
         52,400imes(1.06)5<br>ightarrow52,400imes1.33822552,400 imes (1.06)^5 <br>ightarrow 52,400 imes 1.338225
         - Total Return: $70,118.49

  2. Loan Calculation:
       - Loan Amount: $10,400
       - Interest Rate: 12.7%
       - Compounding Period: Semi-annually
       - Time: 4 years
       - Calculation:
         - Substitute:
         10,400imes(1+rac0.1272)2imes410,400 imes \bigg(1 + rac{0.127}{2}\bigg)^{2 imes 4}
         - Total Payback: $19,844.36

  3. Investment at Compounded Interest:
       - Initial Investment: $5,300
       - Interest Rate: 2.9%
       - Time: 2 years
       - Calculation:
         - Using the formula:
         5,300imes(1.029)25,300 imes (1.029)^2
         - Total Return: $5,588.51

  4. Investment Growth Calculation:
       - Initial Investment: $100
       - Interest Rate: 8.2%
       - Time: 7 years
       - Calculation:
         - Substitute:
         100imes(1.082)7100 imes (1.082)^7
         - Future Value: $192.86

  5. Long-term Investment:
       - Initial Investment: $18,100
       - Interest Rate: 13.6%
       - Compounding Period: Quarterly
       - Time: 7.5 years
       - Calculation:
         - Total =
         18,100imes(1+rac0.1364)4imes7.518,100 imes \bigg(1 + rac{0.136}{4}\bigg)^{4 imes 7.5}
    Total = 18,100×(1+0.1364)4×7.518,100 \times \bigg(1 + \frac{0.136}{4}\bigg)^{4 \times 7.5}

    1. **Identify the values**:
    - Initial Investment, P = $18,100
    - Interest Rate, r = 13.6% = 0.136
    - Compounding Periods per Year = 4 (quarterly)
    - Total Time, t = 7.5 years

    2. **Calculate the effective interest rate per period**:
    - Effective Interest Rate = 0.1364=0.034\frac{0.136}{4} = 0.034

    3. **Calculate the total number of compounding periods**:
    - Total Compounding Periods = 4×7.5=304 \times 7.5 = 30

    4. **Apply the compound interest formula**:
    - Total Amount = 18,100×(1+0.034)3018,100 \times (1 + 0.034)^{30}

    5. **Calculate**:
    - Total Amount = 18,100×(1.034)3018,100 \times (1.034)^{30}
    - First, calculate 1.034301.034^{30}:
    - 1.03430approx2.898281.034^{30} \\approx 2.89828 (using a calculator)
    - Total Amount = 18,100×2.89828approx52,489.1418,100 \times 2.89828 \\approx 52,489.14

    6. **Final result**:
    - **Total Return at the end of 7.5 years**: $52,489.14

Let's verify the calculations step-by-step to ensure accuracy:

  1. Identify the values:
       - Initial Investment, P = $18,100
       - Interest Rate, r = 13.6% = 0.136
       - Compounding Periods per Year = 4 (quarterly)
       - Total Time, t = 7.5 years

  2. Calculate the effective interest rate per period:
       - Effective Interest Rate = 0.1364=0.03440.136​=0.034

  3. Calculate the total number of compounding periods:
       - Total Compounding Periods = 4×7.5=304×7.5=30

  4. Apply the compound interest formula:
       - Total Amount = 18,100×(1+0.034)3018,100×(1+0.034)30

  5. Calculate:
       - Total Amount = 18,100×(1.034)3018,100×(1.034)30
       - Now, calculate 1.034301.03430:
           - 1.03430≈2.898281.03430≈2.89828 (using a calculator)
       - Thus, Total Amount = 18,100×2.89828≈52,489.1418,100×2.89828≈52,489.14

  6. Final result:
       - Total Return at the end of 7.5 years: $52,489.14

  1. Allowance Investment:
       - Initial Amount: $270
       - Interest Rate: 15%
       - Time: 3 years
       - Calculation:
         - 270imes(1.15)3270 imes (1.15)^3

  2. Short Term Loan Return:
       - Loan Amount: $43,000
       - Interest Rate: 3%
       - Time: 2 years
       - Total Return Calculation:
         - 43,000imes(1.03)243,000 imes (1.03)^2

  3. Investment with Semi-Annual Compounding:
       - Initial Investment: $1,200
       - Interest Rate: 5.1%
       - Time: 7.5 years
       - Total Value Calculation:
         - 1,200imes(1+rac0.0512)2imes7.51,200 imes \bigg(1 + rac{0.051}{2}\bigg)^{2 imes 7.5}

  4. Loan Payback Calculation:
       - Loan Amount: $95
       - Interest Rate: 5.2%
       - Time: 1 year
       - Calculation:
         - 95imes(1+rac0.0522)2imes195 imes \bigg(1 + rac{0.052}{2}\bigg)^{2 imes 1}

  5. Investment Total Calculation:
        - Initial Investment: $1,450
        - Interest Rate: 5.4%
        - Time: 6 2/3 years
        - Total Calculation:
          - 1,450imes(1+rac0.05412)12imes6.671,450 imes \bigg(1 + rac{0.054}{12}\bigg)^{12 imes 6.67}

Formulas

  • Volume of a Prism:
      extVolume=ext(Areaofbase)imesext(height)ext{Volume} = ext{(Area of base)} imes ext{(height)}

  • Volume of a Pyramid:
      extVolume=rac13imesext(Areaofbase)imesext(height)ext{Volume} = rac{1}{3} imes ext{(Area of base)} imes ext{(height)}

  • Volume of a Sphere:
      extVolume=rac43imesextπr3ext{Volume} = rac{4}{3} imes ext{π} r^3

  • Surface Area of a Cylinder:
      extSurfaceArea=2extπr2+2extπrhext{Surface Area} = 2 ext{π} r^2 + 2 ext{π} rh

  • Surface Area of a Cone:
      extSurfaceArea=extπr2+extπrsext{Surface Area} = ext{π} r^2 + ext{π} r s

  • Surface Area of a Sphere:
      extSurfaceArea=4extπr2ext{Surface Area} = 4 ext{π} r^2

  • Area of a Circle:
      extArea=extπr2ext{Area} = ext{π} r^2

  • Simple Interest Formula:
      I=PimesrimestI = P imes r imes t

  • Compound Interest Formula:
      A=P(1+i)nA = P(1 + i)^n

  • Surface Area of a Cube:
      extSA=6s2ext{SA} = 6s^2

  • Slope intercept form:
      y=mx+by = mx + b

  • Standard form:
      Ax+By=CAx + By = C

Let's verify the calculations step-by-step to ensure accuracy:

  1. Identify the values:
       - Initial Investment, P = $18,100
       - Interest Rate, r = 13.6% = 0.136
       - Compounding Periods per Year = 4 (quarterly)
       - Total Time, t = 7.5 years

  2. Calculate the effective interest rate per period:
       - Effective Interest Rate = 0.1364=0.034\frac{0.136}{4} = 0.034

  3. Calculate the total number of compounding periods:
       - Total Compounding Periods = 4×7.5=304 \times 7.5 = 30

  4. Apply the compound interest formula:
       - Total Amount = 18,100×(1+0.034)3018,100 \times (1 + 0.034)^{30}

  5. Calculate:
       - Total Amount = 18,100×(1.034)3018,100 \times (1.034)^{30}
       - Now, calculate 1.034301.034^{30}:
           - 1.03430≈2.898281.034^{30} \approx 2.89828 (using a calculator)
       - Thus, Total Amount = 18,100×2.89828≈52,489.1418,100 \times 2.89828 \approx 52,489.14

  6. Final result:
       - Total Return at the end of 7.5 years: $52,489.14