The base of a cylinder is a right triangle topped with a 60 degree sector of a circle, as shown. if the dimensions are in meters and the height of the cylinder is 8 meters, what is the volume of the cylinder

The area of a triangle is given by

... A = (1/2)bh

where b is the length of the base and h is the height perpendicular to the base. For your right triangle, the area is

... A = (1/2)(4 m)(7 m) = 14 m²

The area of a circular sector is given by

... A = (1/2)r²·sin(α)

where r is the radius and α is the angle in radians. For your sector, the area is

... A = (1/2)(4 m)²sin(60°) = 4√3 m²

Then the area of the base of your cylinder is the sum of these, ...

... B = (14 + 4√3) m²

The volume of a cylinder is the product of its base area and height.

... V = Bh

For your cylinder, that volume is

... V = ((14 +4√3) m²)·(8 m)

... V = (112 +32√3) m³ ≈ 167.43 m³

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is