A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #7 #, its base has sides of length #5 #, and its base has a corner with an angle of # pi/3 #. What is the pyramid's surface area?
T S A
AB = BC = CD = DA = a = 5
Height OE = h = 7
OF = a/2 = 5/2 = 2.5
Area of
Lateral surface area
diagonal
#OB = d_2/2 = BCsin (C/2)=5sin(pi/6)= 2.5
#OC = d_1/2 = BC cos (C/2) = 5* cos (pi/6) = 4.33
Area of base ABCD
T S A
T S A
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To find the surface area of the pyramid, we need to calculate the area of each face and then sum them up. The pyramid has four triangular faces and one rhombusshaped base.

Area of the base: Since the base is a rhombus, we can use the formula for the area of a rhombus: A = (diagonal1 * diagonal2) / 2. Given that the side length of the rhombus is 5 and one of its angles is π/3, we can use trigonometry to find the length of the diagonals. The diagonals of a rhombus bisect each other at right angles. Let's denote one of the diagonals as d1 and the other as d2. Using the given side length (5) and angle (π/3), we can find the length of onehalf of one of the diagonals using the cosine function: cos(π/3) = adjacent/hypotenuse. Thus, d1/2 = 5 * cos(π/3) = 5 * (1/2) = 5/2. So, the full diagonal d1 = 2 * (5/2) = 5. Since the diagonals of a rhombus are congruent, d2 = 5 as well. Now, we can calculate the area of the base: A = (5 * 5) / 2 = 25/2.

Area of each triangular face: We can use the formula for the area of a triangle: A = (1/2) * base * height. Given that the base of each triangular face is one side of the rhombus (5) and the height is the height of the pyramid (7), we can calculate the area of each triangular face: A = (1/2) * 5 * 7 = 35/2.

Now, we sum up the areas of all faces: There are four triangular faces, each with an area of 35/2, and one rhombusshaped base with an area of 25/2. Total surface area = 4 * (35/2) + 25/2 = 140/2 + 25/2 = 165/2.
So, the surface area of the pyramid is 165/2 square units.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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