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Answer :
Based on the available information, the maximum number of spheres the truck can carry is approximately 127.
To find out the maximum number of spheres the truck can carry, calculate the volume of each sphere and then divide the maximum load weight limit by the weight of each sphere.
1. Calculate the volume of each sphere:
- The formula for the volume of a sphere is:
[tex]( V = \frac{4}{3} \pi r^3 )[/tex]
, where r is the radius of the sphere.
- Given the diameter is 22 inches, the radius
[tex]( r = \frac{22}{2} = 11 ) inches.[/tex]
- Convert the diameter to feet:
[tex] 22 \text{ inches} \div 12 = 1.8333 \text{ feet}.[/tex]
- So, the radius in feet is
[tex]( 11 \text{ inches} \div 12 = 0.9167 \text{ feet})[/tex]
- Now, calculate the volume:
[tex] \[ V = \frac{4}{3} \pi (0.9167)^3 \][/tex]
2. Calculate the weight of each sphere:
- The weight of each sphere is its volume multiplied by the weight per cubic foot of concrete.
- Weight of each sphere = Volume of sphere * Weight per cubic foot of concrete
- Weight of each sphere =
[tex] ( V \times 137 )[/tex]
pounds
3. Calculate the maximum number of spheres the truck can carry:
- Given that the maximum load weight limit of the truck is 6.0 short tons, convert this to pounds (1 short ton = 2000 pounds).
- Maximum load weight limit = 6.0 short tons * 2000 pounds/short ton
Now, calculate:
1. Volume of each sphere:
[tex] \[ V = \frac{4}{3} \pi (0.9167)^3 \][/tex]
2. Weight of each sphere:
[tex] \[ \text{Weight of each sphere} = V \times 137 \][/tex]
3. Maximum load weight limit of the truck:
[tex] \[ \text{Maximum load weight limit} = 6.0 \times 2000 \][/tex]
4. Maximum number of spheres the truck can carry:
[tex] \[ \text{Maximum number of spheres} = \frac{\text{Maximum load weight limit}}{\text{Weight of each sphere}} \][/tex]
Calculate:
1. Volume of each sphere:
[tex] \[ V = \frac{4}{3} \pi (0.9167)^3 \][/tex]
[tex] \[ V ≈ 0.685 \text{ cubic feet} \][/tex]
2. Weight of each sphere:
[tex]\[ \text{Weight of each sphere} ≈ 0.685 \times[/tex]
[tex]137 ≈ 93.89 \text{ pounds} \][/tex]
3. Maximum load weight limit of the truck:
[tex] \[ \text{Maximum load weight limit} = 6.0 \times[/tex]
[tex]2000 = 12000 \text{ pounds} \][/tex]
4. Maximum number of spheres the truck can carry: Maximum number of spheres} = 12000/93.89 ≈ 127.86
Therefore, the maximum number of spheres the truck can carry is approximately 127 (rounded down).
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