Solve the equation:a right pyramid with vertical height stand on rectangular base widith that measures 20cm by 14cm the hieght of the pyramid is 10cm calculate to 3 significant figure the length of the slant height , the surface area of the pyramid, volume of the pyramid
To solve the given problem, we need to use the properties of a right pyramid and perform some calculations based on the given dimensions. Let's break it down step-by-step:
Step 1: Calculate the length of the slant height (l):
The slant height (l) can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (l) is equal to the sum of the squares of the other two sides:
Base length (b) = 20 cm
Base width (w) = 14 cm
Height (h) = 10 cm
Using the Pythagorean theorem:
l^2 = b^2 + w^2
Substituting the given values:
l^2 = 20^2 + 14^2
l^2 = 400 + 196
l^2 = 596
Taking the square root of both sides to solve for l:
l = √596
l ≈ 24.4 cm (rounded to 3 significant figures)
Therefore, the length of the slant height is approximately 24.4 cm.
Step 2: Calculate the surface area of the pyramid:
To find the surface area of a pyramid, we need to calculate the area of the base and the area of the lateral faces separately and sum them up.
The area of the rectangular base (A_base) is given by:
A_base = length × width
A_base = 20 cm × 14 cm
A_base = 280 cm²
The area of each lateral face is a triangle, and the area of a triangle can be calculated by multiplying half of the base length by the corresponding height. In this case, the base length and height of each lateral face are given by the base length (20 cm) and the slant height (24.4 cm), respectively.
The area of each lateral face (A_lateral) is given by:
A_lateral = (1/2) × base length × slant height
A_lateral = (1/2) × 20 cm × 24.4 cm
A_lateral = 244 cm²
Since there are four lateral faces on a pyramid, the total area of the lateral faces (A_total_lateral) is:
A_total_lateral = 4 × A_lateral
A_total_lateral = 4 × 244 cm²
A_total_lateral = 976 cm²
Finally, the surface area (A_total) of the pyramid is the sum of the base area (A_base) and the total area of the lateral faces (A_total_lateral):
A_total = A_base + A_total_lateral
A_total = 280 cm² + 976 cm²
A_total = 1256 cm²
Therefore, the surface area of the pyramid is 1256 cm².
Step 3: Calculate the volume of the pyramid:
The volume of a pyramid can be calculated by multiplying the area of the base by the height and dividing the result by 3:
The volume of the pyramid (V) is given by:
V = (1/3) × base area × height
V = (1/3) × A_base × h
V = (1/3) × 280 cm² × 10 cm
V ≈ 933.3 cm³ (rounded to 3 significant figures)
Therefore, the volume of the pyramid is approximately 933.3 cm³.