This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Determine the dimensions of the rectangular base and the height of the pyramid. The base ABCD is a rectangle with AD = 12 cm and AB = 16 cm. Since it's a rectangle, BC = AD = 12 cm. O is the center of the base, and VO is the vertical height of the pyramid. First, find the length of the diagonal AC of the base using the Pythagorean theorem in ABC: AC^2 = AB^2 + BC^2 AC^2 = 16^2 + 12^2 AC^2 = 256 + 144 AC^2 = 400 AC = sqrt(400) = 20 cm Since O is the center of the rectangle, OC is half of AC: OC = (1)/(2) AC = (1)/(2)(20) = 10 cm Now, find the vertical height VO using the Pythagorean theorem in VOC: VC^2 = VO^2 + OC^2 Given VC = 26 cm: 26^2 = VO^2 + 10^2 676 = VO^2 + 100 VO^2 = 676 - 100 VO^2 = 576 VO = sqrt(576) = 24 cm Step 2: Identify the angle between the planes ADCB (base) and VAB. The line of intersection of the two planes is AB. To find the angle between the planes, we need to find a line in each plane that is perpendicular to the line of intersection (AB) at the same point. Let M be the midpoint of AB. In the base plane ADCB, the line segment OM is perpendicular to AB (since O is the center and M is the midpoint of a side, OM is parallel to AD and BC, which are perpendicular to AB). The length of OM is half the length of AD: OM = (1)/(2) AD = (1)/(2)(12) = 6 cm In the plane VAB, the line segment VM is the slant height of the face VAB. Since VAB is an isosceles triangle (VA = VB in a right pyramid), VM is perpendicular to AB. Therefore, the angle between the planes ADCB and VAB is VMO. Step 3: Calculate the angle VMO. Consider the right-angled triangle VMO. We have: Opposite side to VMO: VO = 24 cm Adjacent side to VMO: OM = 6 cm Using the tangent function: ( VMO) = OppositeAdjacent = (VO)/(OM) ( VMO) = (24)/(6) ( VMO) = 4 VMO = (4) VMO ≈ 75.9637^ Rounding to one decimal place: VMO ≈ 76.0^ The angle between the planes ADCB and VAB is 76.0^. 3 done, 2 left today. You're making progress.