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.
To find the three face angles at A, we can use the properties of a pyramid.
Step 1: Let's start by finding the length of AC. Since AB and BC are given, we can use the Pythagorean theorem to find AC.
AC = √(AB² + BC²)
Substituting the given values, we get:
AC = √(9² + 12²) = √(81 + 144) = √225 = 15 in
Step 2: Now, let's find the length of AD. Since BD is given, we can use the Pythagorean theorem again.
AD = √(AB² - BD²)
Substituting the given values, we get:
AD = √(9² - 5²) = √(81 - 25) = √56 ≈ 7.48 in
Step 3: We can now find the angles at A using the law of cosines. Let's call the angles at A α, β, and γ.
cos(α) = (AC² + AD² - CD²) / (2 * AC * AD)
cos(β) = (AB² + AD² - BD²) / (2 * AB * AD)
cos(γ) = (AC² + AB² - BC²) / (2 * AC * AB)
Substituting the given lengths, we get:
cos(α) = (15² + 7.48² - 5²) / (2 * 15 * 7.48)
cos(β) = (9² + 7.48² - 5²) / (2 * 9 * 7.48)
cos(γ) = (15² + 9² - 12²) / (2 * 15 * 9)
Calculating these values:
cos(α) ≈ 0.894
cos(β) ≈ 0.999
cos(γ) ≈ 0.559
To find the angles α, β, and γ, we can use the inverse cosine function:
α = cos^(-1)(0.894) ≈ 26.57 degrees
β = cos^(-1)(0.999) ≈ 9.69 degrees
γ = cos^(-1)(0.559) ≈ 55.74 degrees
Therefore, the three face angles at A are approximately 26.57 degrees, 9.69 degrees, and 55.74 degrees.
To calculate the total surface area of the solid, we need to find the areas of each face and then sum them up.
Step 4: The area of each face can be found using the formula for the area of a triangle, which is 1/2 * base * height. Since all the triangles in the pyramid have a right angle at B, the base and height of each triangle are AB and BC, respectively.
Area of each face = 1/2 * AB * BC
Substituting the given values:
Area of each face = 1/2 * 9 * 12 = 54 square inches
Step 5: The total surface area of the solid is the sum of the areas of all four faces.
Total surface area = 4 * Area of each face
= 4 * 54 square inches
= 216 square inches
Therefore, the total surface area of the solid is 216 square inches.