An Engineer is designing a bolt to fit a new machine. The bolt was the shape of a hexagon with a cymetrical hole in its center. The greatest distance across the bolt is 1.25cm, and the depth of 0.5cm. The diameter of the whole is 0.75cm while each side of the bolt has a length of 0.625. The area of the base of a hexogonal solid with these dimensions, withought the whole, is given by the formula A= 0.5(0.54), where p is the perimeter of the hexagon. What volume of the meter to the nearest hundreth cubic centimeters. will be required to make the bolt
To find the volume of the bolt, we need to calculate the volume of the hexagonal solid (without the hole) and then subtract the volume of the hole.
Step 1: Calculate the area of the base of the hexagonal solid.
The formula given in the question is A = 0.5 * (0.54) * p, where p is the perimeter of the hexagon. The perimeter of a hexagon is calculated by multiplying the length of one side by 6 (since a hexagon has 6 sides).
Given that each side of the bolt has a length of 0.625 cm, the perimeter becomes p = 0.625 * 6 = 3.75 cm.
Substituting the value of p into the formula, we get A = 0.5 * (0.54) * 3.75.
Step 2: Calculate the area of the hole.
The hole in the center of the bolt is circular, with a diameter of 0.75 cm. Using the formula for the area of a circle (A = π * r^2) where r is the radius (half of the diameter), we can calculate the area of the hole.
The radius (r) of the hole is 0.75 / 2 = 0.375 cm.
Substituting the value of r into the formula, we get A = π * (0.375)^2.
Step 3: Calculate the volume.
The volume of the hexagonal solid can be calculated by multiplying the area of the base (A) with the depth of the bolt.
Given that the depth is 0.5 cm, the volume of the hexagonal solid is V_solid = A * depth.
Similarly, the volume of the hole can be calculated by multiplying the area of the hole (A_hole) with the depth of the bolt.
Given that the depth is 0.5 cm, the volume of the hole is V_hole = A_hole * depth.
Finally, the volume of the bolt is obtained by subtracting the volume of the hole from the volume of the solid:
V_bolt = V_solid - V_hole.
Now, use the formulas and substitute the calculated values to find the volume of the bolt.
To find the volume of the bolt, we first need to find the area of the base of the hexagonal solid.
Given:
Perimeter of the hexagon (p) = 6 * side length of the hexagon = 6 * 0.625 = 3.75 cm
Using the formula given:
Area of the base (A) = 0.5 * p * apothem (where apothem is the distance from the center of the hexagon to the midpoint of any side)
The apothem can be found by dividing the distance across the bolt (1.25 cm) by 2:
Apothem = 1.25 cm / 2 = 0.625 cm
Now we can calculate the area of the base:
A = 0.5 * 3.75 cm * 0.625 cm = 1.171875 cm²
Since the depth of the bolt is 0.5 cm, the volume is given by multiplying the area of the base by the depth:
Volume = 1.171875 cm² * 0.5 cm = 0.5859375 cm³
Rounding to the nearest hundredth, the volume of the bolt required is approximately 0.59 cm³.