An object is made of a prism with a spherical cap on its square shaped top. The cap's base has a diameter equal to the lengths of the top. The prism's height is # 15 #, the cap's height is #8 #, and the cap's radius is #9 #. What is the object's volume?

Answer 1

Total volume#=pi(-4root2(5)-162*arcsin4root2(5)/9+82*4root2(5)-(4root2(5))^3/3)+4800#

Let' consider a circumference with centre in #C(0;-1)# and radius 9. Its equation is #x^2+(y+1)^2=81# and its vertical diameter is 8 long in the positive y semiplane (for #y> 0#) just the height of our cap. The intersect #OI# of it with the x-axys is the height of the rectangle triangle inscripted inside the vertical semicircunference where the projection of the two catheti are just 8 and 10. As a result, according to the second euclid's theorem #IO=root2(80)=4root2(5)#.
Notice that #2*OI# is the side of the prism's base, i.e. #L=8root2(5)# whereas its height is #H=15#, so the prism's volume is #V_(prism)=L^2*H=(8*root2(5))^2*15=4800 #
Moving to the cap's volume, it can be seen as the volume of the solid generated by the arc of circumference in the first quarter of the cartesian axis, whose explicit equation is #y=root2(81-x^2)-1# for #0 < x<4root2(5)#
It results from the second theorem of Guldino that #V_("cap")=2pi*x_("barycentre")*area_("section")# #V_("cap")= pi*(int_0^(4root2(5))(root2(81-x^2)-1)^2dx)/(int_0^(4root2(5))dx)*int_0^(4root2(5))dx# that can be simplified as #V_("cap")=pi*int_0^(4root2(5))(-2root2(81-x^2)+1+81-x^2)dx#

solved the integral that we obtain

#V_("cap")=pi(-2*(1/2xroot2(81-x^2)+81*arcsin(x/9))+82x-x^3/3)|_0^(4root2(5))#
#V_("cap")=pi(-4root2(5)-162*arcsin4root2(5)/9+82*4root2(5)-(4root2(5))^3/3)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the volume of the object, you calculate the volume of the prism and the volume of the spherical cap separately, then add them together.

The volume of the prism is given by V_prism = base area * height_prism. Since the prism's base is square-shaped with side length equal to the diameter of the cap's base, the base area is (2 * 9)^2 = 324 square units. Given the prism's height is 15 units, the volume of the prism is V_prism = 324 * 15 = 4860 cubic units.

The volume of the spherical cap is given by V_cap = (1/3) * π * h_cap^2 * (3R - h_cap), where h_cap is the height of the cap and R is the radius of the sphere from which the cap is cut. Given the cap's height is 8 units and its radius is 9 units, the volume of the cap is V_cap = (1/3) * π * 8^2 * (3*9 - 8) = (64/3) * π * (27 - 8) = (64/3) * π * 19 = 4057.33 cubic units (approximately).

Adding the volumes of the prism and the cap together gives the total volume of the object:

Total Volume = V_prism + V_cap = 4860 + 4057.33 ≈ 8917.33 cubic units.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7