Find the surface area of each figure to the nearest tenth. show your work.

Let us name the figures from left to right as Figures 1, 2, 3, and 4.

Figure 1: Cylinder

The formula for surface area of cylinder is:

SA = 2 π r h

where r = 5 cm  and        h = 10 cm

SA = 2 π (5 cm) (10 cm)

SA = 100 π cm^2 = 314.16 cm^2

Figure 2: Triangular prism

SA = (a + b + c) h

where a, b, c are the sides of the triangle and h is the length of the prism

a = 4 m

b = c = 2.5 m

h = 6 m

SA = (4 m + 2.5 m + 2.5 m) 6 m

SA = 54 m^2

Figure 3: Rectangular prism

SA = 2lw + 2wh + 2lh

where l, w and h are the sides

l = 3.5 ft

w = 2.4 ft

h = 5.2 ft

SA = 2 (3.5*2.4 + 2.4*5.2 + 3.5*5.2)

SA = 78.16 ft^2

Figure 4: Rectangular prism

SA = 2lw + 2wh + 2lh

where l, w and h are the sides

l = 6 ft

w = 4.5 ft

h =12 ft

SA = 2 (6*4.5 + 4.5*12 + 6*12)

SA = 306 ft^2

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answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.