Answer:
The box has a total volume of 36 cubic inches.
Step-by-step explanation:
The total volume of the box is given by the volume that is filled with books plus the volume that doesn't have any books in it. Therefore, in order to solve this problem we must sum the two volumes. We have:
total volume = volume with books + volume without books
total volume = 24 + 12
total volume = 36 cubic inches
The box has a total volume of 36 cubic inches.
Answer:
1.57 radians
Step-by-step explanation:
Divide both sides by 360, you get 1° = π/180. This means 70° is equal to (70)(π/180). Then simplify this, and 70° = (7π)/18 radians, or approximately 1.221730476396 radians. This seems reasonable because 90°, is a quarter circle and (1/4) of 2π is π/2, which is approximately 1.57 radians.
Answer:
π/6 [37^(³/₂) − 1] ≈ 117.3187
Step-by-step explanation:
The surface area is:
S = ∫ 2π (x − 0) √(1 + (dx/dy)²) dy
0 ≤ x ≤ 3, so -4 ≤ y ≤ 5.
Find dx/dy.
y = 5 − x²
x² = 5 − y
x = √(5 − y)
dx/dy = ½ (5 − y)^(-½) (-1)
dx/dy = -½ (5 − y)^(-½)
(dx/dy)² = ¼ (5 − y)^(-1)
(dx/dy)² = 1 / (4 (5 − y))
Plug in:
S = ∫₋₄⁵ 2π x √(1 + 1 / (20 − 4y)) dy
S = ∫₋₄⁵ 2π √(5 − y) √(1 + 1 / (4 (5 − y))) dy
S = ∫₋₄⁵ 2π √((5 − y) + 1/4)) dy
S = ∫₋₄⁵ 2π √(5.25 − y) dy
If u = 5.25 − y, then du = -dy.
S = ∫ 2π √u (-du)
S = -2π ∫ √u du
S = -2π (⅔ u^(³/₂))
S = -4π/3 u^(³/₂)
Substitute back:
S = -4π/3 (5.25 − y)^(³/₂)
Evaluate between y=-4 and y=5.
S = [-4π/3 (5.25 − 5)^(³/₂)] − [-4π/3 (5.25 − -4)^(³/₂)]
S = -4π/3 (0.25)^(³/₂) + 4π/3 (9.25)^(³/₂)
S = π/6 [37^(³/₂) − 1]
S ≈ 117.3187
