: in the pyramid shown in the figure , AB=9 in ., BC= 12in ,and BD= 5 in . the three face angle at B are each 90 degree . calculate the three face angles at A and the total surface of the solid.
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To calculate the three face angles at point A and the total surface area of the pyramid, we need to use the Pythagorean theorem and trigonometric ratios. Let's break down the steps:
Step 1: Find the length of AC using the Pythagorean theorem.
Since triangle ABC is a right triangle with a 90-degree angle at B, we can use the Pythagorean theorem to find AC.
Applying the theorem: AC^2 = AB^2 + BC^2
Substituting the given values: AC^2 = 9^2 + 12^2
Simplifying: AC^2 = 81 + 144
AC^2 = 225
Taking the square root of both sides: AC = √225 = 15 in
Step 2: Calculate the length of AD using the Pythagorean theorem.
Triangle ABD is also a right triangle with a 90-degree angle at B.
Applying the theorem: AD^2 = AB^2 - BD^2
Substituting the given values: AD^2 = 9^2 - 5^2
Simplifying: AD^2 = 81 - 25
AD^2 = 56
Taking the square root of both sides: AD = √56 = 4√14 in
Step 3: Calculate the length of CD.
Since AC is the hypotenuse of triangle ADC, we can use the Pythagorean theorem again:
CD^2 = AC^2 - AD^2
Substituting the known values: CD^2 = 15^2 - (4√14)^2
Simplifying: CD^2 = 225 - 16 * 14
CD^2 = 225 - 224
CD = 1 in
Step 4: Calculate the face angles at point A.
To find the face angles at A, we can use trigonometric ratios. Since we know all the side lengths in triangle ABC, we can use the inverse trigonometric functions.
Using the inverse tangent (arctan) function: tan(A) = BC / AB
Substituting the known values: tan(A) = 12 / 9
Simplifying: tan(A) = 4 / 3
Taking the inverse tangent of both sides: A = arctan(4 / 3)
Using a calculator or a trigonometric table, we find A ≈ 53.13 degrees.
Since the three face angles at A are the same, each angle is approximately 53.13 degrees.
Step 5: Calculate the total surface area of the pyramid.
The total surface area of a pyramid can be calculated by summing the areas of all its faces. In this case, we have four triangular faces and one rectangular face.
Rectangular face ADCE:
The area of a rectangle is given by length × width. The length is AD (4√14 in) and the width is CD (1 in):
Area = AD × CD = (4√14)(1) = 4√14 in^2.
Triangular faces ABD, ACD, and ABC:
The area of a triangle can be calculated using Heron's formula or by using one side length and the height of the triangle. In this case, we have the base length AB (9 in) and the height.
The height can be found using the Pythagorean theorem: height = √(AC^2 - AD^2) = √(225 - 56) = √169 = 13 in.
Therefore, the area of each triangular face is: (1/2) × base × height = (1/2)(9)(13) = 58.5 in^2.
Since there are three identical triangular faces, the total area of the triangular faces is 3 × 58.5 = 175.5 in^2.
Adding the area of the rectangular face to the total area of the triangular faces:
Total surface area = 175.5 + 4√14 ≈ 177.02 + 4√14 in^2.
So, the three face angles at A are approximately 53.13 degrees, and the total surface area of the pyramid is approximately 177.02 + 4√14 square inches.