a rectangular base that is 300 cm long and 110 cm wide. The front and back faces are vertical, but the left and right faces are slanted at 30° from the vertical and measure 110 cm by 170 cm.

Question

A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is 300 cm long and 110 cm wide. The front and back faces are vertical, but the left and right faces are slanted at 30° from the vertical and measure 110 cm by 170 cm.

(a) Let u, v, and w be the three vectors shown in the figure. Find u · (v ⨯ w). [Hint: Recall that u · v = |u||v| cos(𝜃) and |u ⨯ v| = |u||v| sin(𝜃).]

(b) What is the capacity of the tank in liters? [Note: 1 L = 1000 cm³.] (Round your answer to the nearest liter.)

Answer 1

(a) IuI = 110cm; Iv×wI = IvI·IwIsinθ = 170·300sin60° = 25500√3 cm²

u·(v×w) = 110·25500√3scos0° = 2805000√3 cm³·

(b) Volume Q = 300·110·170cos30° = 2805000√3 cm³ = 2805√3 L = 4858.4 L