The rectangular box shown in the figure has dimensions 10" × 7" × 3" (a = 10, b = 7, c = 3). Approximate the angle θ formed by a diagonal of the base and a diagonal of the 7" × 3" side. (Round your answer to two decimal places.)
That is the angle between the vectors (7,10,0) and (7,0,3)
So since
(7,10,0)•(7,0,3) = √149√58 cosθ,
cosθ = 49/√8642 = 0.527
To approximate the angle θ, we can use the formula for the cosine of an angle in a right triangle:
cos(θ) = adjacent side / hypotenuse
In this case, we are interested in the angle formed by the diagonal of the base (hypotenuse) and the diagonal of the 7" × 3" side (adjacent side).
First, let's find the length of the hypotenuse (diagonal of the base) using the Pythagorean theorem:
h = √(a² + b²)
h = √(10² + 7²)
h = √(100 + 49)
h = √149
h ≈ 12.21
Next, let's find the length of the adjacent side (diagonal of the 7" × 3" side) using the Pythagorean theorem:
a' = √(b² + c²)
a' = √(7² + 3²)
a' = √(49 + 9)
a' = √58
a' ≈ 7.62
Now we have the lengths of the adjacent side (a') and the hypotenuse (h) of the right triangle formed by the diagonals. We can use these values to find the cosine of the angle θ:
cos(θ) = a' / h
cos(θ) = 7.62 / 12.21
cos(θ) ≈ 0.625
Finally, we can find the value of θ by taking the arccosine (inverse cosine) of the calculated value:
θ ≈ arccos(0.625)
θ ≈ 51.32 degrees
Therefore, the approximate angle θ formed by a diagonal of the base and a diagonal of the 7" × 3" side is approximately 51.32 degrees.