A rectangular box shown has dimensions 10" multiplied by 6" multiplied by 4" (a = 10, b = 6, c = 4). Approximate the angle theta formed by a diagonal of the base and a diagonal of the 6" multiplied by 4" side. (Give the answer to two decimal places.)
To find the angle theta formed by the diagonal of the base and a diagonal of one of the sides, we can use the properties of right triangles.
First, let's find the length of the diagonal of the base (d1):
Using the Pythagorean theorem, we have:
d1² = 10² + 6²
d1² = 100 + 36
d1² = 136
d1 ≈ √136
d1 ≈ 11.66 inches
Next, let's find the length of the diagonal of the 6" multiplied by 4" side (d2):
Using the Pythagorean theorem again, we have:
d2² = 6² + 4²
d2² = 36 + 16
d2² = 52
d2 ≈ √52
d2 ≈ 7.21 inches
Now, we can find the angle theta using the inverse tangent function (tan⁻¹):
θ = tan⁻¹(d2 / d1)
θ = tan⁻¹(7.21 / 11.66)
θ ≈ 32.80 degrees
Therefore, the approximate angle theta (θ) formed by the diagonal of the base and a diagonal of the 6" multiplied by 4" side is approximately 32.80 degrees.