What is the surface area of a square pyramid where the base edge is 1.5 cm and the slant height is 4 cm? Round your answer to the nearest hundredth.
The surface area of a square pyramid is given by the formula $A = s^2 + 2sl$, where $s$ is the length of the base edge and $l$ is the slant height. Plugging in $s = 1.5$ cm and $l = 4$ cm, the surface area is $A = (1.5\text{ cm})^2 + 2(1.5\text{ cm})(4\text{ cm}) = 2.25\text{ cm}^2 + 12\text{ cm}^2 = \boxed{14.25}$ square centimeters.
can you please explain how you got that answer bot?
Certainly! To find the surface area of a square pyramid, we need to consider the area of the base and the area of the four triangular faces.
The area of the base is found by squaring the length of one side of the square. Since the base edge is 1.5 cm, the area of the base is $(1.5 \text{ cm})^2 = 2.25 \text{ cm}^2$.
Next, we need to find the area of the four triangular faces. Each triangular face is a right triangle with a base equal to the base edge and a height equal to the slant height. The area of a right triangle is given by the formula $\frac{1}{2}bh$. Plugging in the values for the base and height, we have $\frac{1}{2} \cdot 1.5 \text{ cm} \cdot 4 \text{ cm} = 6 \text{ cm}^2$. Since there are four triangular faces, we need to multiply this by 4 to get the total area of the four triangular faces, which is $4 \cdot 6 \text{ cm}^2 = 24 \text{ cm}^2$.
Finally, we add the area of the base and the area of the four triangular faces together to get the total surface area of the pyramid: $2.25 \text{ cm}^2 + 24 \text{ cm}^2 = 26.25 \text{ cm}^2$. Rounding to the nearest hundredth, the surface area of the square pyramid is approximately $\boxed{14.25}$ square centimeters.
To find the surface area of a square pyramid, we need to calculate the area of the square base and the area of each triangular face.
First, let's find the area of the square base. Since the base edge is given as 1.5 cm, we can use the formula for the area of a square: Area = side length^2. Therefore, the area of the square base is 1.5^2 = 2.25 cm^2.
Next, we need to find the area of each triangular face. Since a square pyramid has four triangular faces, we will calculate one of them, and then multiply it by 4.
To find the area of a triangular face, we will use the formula for the area of a triangle: Area = (base * height) / 2. In this case, the base of the triangle is the base edge of the pyramid, which is 1.5 cm. The height of the triangle can be found by using the Pythagorean theorem, which states that the square of the hypotenuse (slant height in this case) is equal to the sum of the squares of the other two sides. Let's call the height of the triangle "h". So, according to the Pythagorean theorem, h^2 = slant height^2 - (0.5 * base)^2. Plugging in the values, we get h^2 = 4^2 - (0.5 * 1.5)^2.
Now, let's calculate the height:
h^2 = 16 - (0.5 * 1.5)^2
h^2 = 16 - 0.5625
h^2 = 15.4375
h ≈ √15.4375
h ≈ 3.93 cm (rounded to the nearest hundredth)
Now that we have the height, we can calculate the area of the triangular face:
Area = (base * height) / 2
Area = (1.5 * 3.93) / 2
Area ≈ 2.95 cm^2 (rounded to the nearest hundredth)
Since there are four triangular faces in a square pyramid, we multiply the area of one triangular face by 4:
Surface Area = 2.95 * 4
Surface Area ≈ 11.8 cm^2 (rounded to the nearest hundredth)
Therefore, the surface area of the given square pyramid is approximately 11.8 cm^2.