Answer :
Alright, let's solve this step by step.
Step 1: Convert the dimensions to meters.
The given dimensions are in centimeters. We need to convert them to meters because the metal cost is given per square meter.
- Diameter: 7 cm [tex]\( (0.07 \; m) \)[/tex]
- Height: 21 cm [tex]\( (0.21 \; m) \)[/tex]
Step 2: Calculate the radius of the cylinder.
The radius [tex]\( r \)[/tex] is half the diameter.
[tex]\[ r = \frac{diameter}{2} = \frac{0.07 \; m}{2} = 0.035 \; m \][/tex]
Step 3: Calculate the surface area of the side of the cylinder.
The lateral surface area [tex]\( A_{side} \)[/tex] of a cylinder can be found by the formula:
[tex]\[ A_{side} = 2 \pi r h \][/tex]
Where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
[tex]\[ A_{side} = 2 \pi \times 0.035 \; m \times 0.21 \; m \][/tex]
[tex]\[ A_{side} \approx 2 \pi \times 0.035 \times 0.21 \][/tex]
[tex]\[ A_{side} \approx 2 \pi \times 0.00735 \][/tex]
[tex]\[ A_{side} \approx 2 \times 3.14159 \times 0.00735 \][/tex]
[tex]\[ A_{side} \approx 0.0461843 \; m^2 \][/tex]
Step 4: Calculate the surface area of the base of the cylinder.
The surface area [tex]\( A_{base} \)[/tex] of the base (which is a circle) can be found by the formula:
[tex]\[ A_{base} = \pi r^2 \][/tex]
[tex]\[ A_{base} = \pi \times (0.035 \; m)^2 \][/tex]
[tex]\[ A_{base} = \pi \times 0.001225 \; m^2 \][/tex]
[tex]\[ A_{base} = 3.14159 \times 0.001225 \; m^2 \][/tex]
[tex]\[ A_{base} \approx 0.00384849 \; m^2 \][/tex]
Step 5: Calculate the total surface area of a single cylinder.
Because the container is open at the top, the total surface area [tex]\( A_{total-single} \)[/tex] will be the sum of the side area and the base area.
[tex]\[ A_{total-single} = A_{side} + A_{base} \][/tex]
[tex]\[ A_{total-single} \approx 0.0461843 \; m^2 + 0.00384849 \; m^2 \][/tex]
[tex]\[ A_{total-single} \approx 0.05003279 \; m^2 \][/tex]
Step 6: Find the total area for 125 cylinders.
Now we have the total area for one cylinder, we can calculate the area for all 125 cylinders by multiplying the single cylinder area by 125.
[tex]\[ A_{total-all} = A_{total-single} \times 125 \][/tex]
[tex]\[ A_{total-all} \approx 0.05003279 \; m^2 \times 125 \][/tex]
[tex]\[ A_{total-all} \approx 6.25409875 \; m^2 \][/tex]
Step 7: Calculate the cost for the metal for all cylinders.
Finally, we find the total cost by multiplying the total area by the cost per square meter.
[tex]\[ Cost = A_{total-all} \times Cost_{per-square-meter} \][/tex]
[tex]\[ Cost \approx 6.25409875 \; m^2 \times \$4.50 \][/tex]
[tex]\[ Cost \approx \$28.143444375 \][/tex]
Step 8: Round to the nearest cent if necessary.
Since we're dealing with money, we should round to the nearest cent.
[tex]\[ Cost \approx \$28.14 \][/tex]
Therefore, the cost of making 125 cylindrical tennis ball containers with the given dimensions and metal cost is approximately $28.14.
Step 1: Convert the dimensions to meters.
The given dimensions are in centimeters. We need to convert them to meters because the metal cost is given per square meter.
- Diameter: 7 cm [tex]\( (0.07 \; m) \)[/tex]
- Height: 21 cm [tex]\( (0.21 \; m) \)[/tex]
Step 2: Calculate the radius of the cylinder.
The radius [tex]\( r \)[/tex] is half the diameter.
[tex]\[ r = \frac{diameter}{2} = \frac{0.07 \; m}{2} = 0.035 \; m \][/tex]
Step 3: Calculate the surface area of the side of the cylinder.
The lateral surface area [tex]\( A_{side} \)[/tex] of a cylinder can be found by the formula:
[tex]\[ A_{side} = 2 \pi r h \][/tex]
Where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
[tex]\[ A_{side} = 2 \pi \times 0.035 \; m \times 0.21 \; m \][/tex]
[tex]\[ A_{side} \approx 2 \pi \times 0.035 \times 0.21 \][/tex]
[tex]\[ A_{side} \approx 2 \pi \times 0.00735 \][/tex]
[tex]\[ A_{side} \approx 2 \times 3.14159 \times 0.00735 \][/tex]
[tex]\[ A_{side} \approx 0.0461843 \; m^2 \][/tex]
Step 4: Calculate the surface area of the base of the cylinder.
The surface area [tex]\( A_{base} \)[/tex] of the base (which is a circle) can be found by the formula:
[tex]\[ A_{base} = \pi r^2 \][/tex]
[tex]\[ A_{base} = \pi \times (0.035 \; m)^2 \][/tex]
[tex]\[ A_{base} = \pi \times 0.001225 \; m^2 \][/tex]
[tex]\[ A_{base} = 3.14159 \times 0.001225 \; m^2 \][/tex]
[tex]\[ A_{base} \approx 0.00384849 \; m^2 \][/tex]
Step 5: Calculate the total surface area of a single cylinder.
Because the container is open at the top, the total surface area [tex]\( A_{total-single} \)[/tex] will be the sum of the side area and the base area.
[tex]\[ A_{total-single} = A_{side} + A_{base} \][/tex]
[tex]\[ A_{total-single} \approx 0.0461843 \; m^2 + 0.00384849 \; m^2 \][/tex]
[tex]\[ A_{total-single} \approx 0.05003279 \; m^2 \][/tex]
Step 6: Find the total area for 125 cylinders.
Now we have the total area for one cylinder, we can calculate the area for all 125 cylinders by multiplying the single cylinder area by 125.
[tex]\[ A_{total-all} = A_{total-single} \times 125 \][/tex]
[tex]\[ A_{total-all} \approx 0.05003279 \; m^2 \times 125 \][/tex]
[tex]\[ A_{total-all} \approx 6.25409875 \; m^2 \][/tex]
Step 7: Calculate the cost for the metal for all cylinders.
Finally, we find the total cost by multiplying the total area by the cost per square meter.
[tex]\[ Cost = A_{total-all} \times Cost_{per-square-meter} \][/tex]
[tex]\[ Cost \approx 6.25409875 \; m^2 \times \$4.50 \][/tex]
[tex]\[ Cost \approx \$28.143444375 \][/tex]
Step 8: Round to the nearest cent if necessary.
Since we're dealing with money, we should round to the nearest cent.
[tex]\[ Cost \approx \$28.14 \][/tex]
Therefore, the cost of making 125 cylindrical tennis ball containers with the given dimensions and metal cost is approximately $28.14.
