Solve the problem. The surface area of a square pyramid is 116 in.2 and the total area of the pyramid’s four triangular faces is 80 in.2
What is the length of one of the sides? Show your work.(2 points)
Let's start by using the formula for the surface area of a square pyramid:
Surface area = base area + 4 × (area of triangular face)
Since we know the surface area and the area of the triangular faces, we can substitute and solve for the base area:
116 in.2 = base area + 4 × 80 in.2/4
116 in.2 = base area + 80 in.2
Base area = 116 in.2 - 80 in.2
Base area = 36 in.2
Since the base of a square pyramid is a square, we can find the length of one of its sides by taking the square root of its area:
Base area = side2
36 in.2 = side2
Side = √36 in.
Side = 6 in.
Therefore, the length of one of the sides of the square base is 6 inches.
Well, it seems like this square pyramid is hiding some secrets! Let's uncover its side length with a bit of humor, shall we?
Let's imagine the sides of the square pyramid are the legs of a comedy duo. They are always together and they must have a funny dynamic!
So, let's call one side of the square pyramid "Larry" and the other side "Curly". These two sides, Larry and Curly, will form a right triangle with the height of the pyramid as the hypotenuse.
But here's the twist: the area of this right triangle is 80 in²!
Now, comedy comes in many shapes and sizes, and the formula for the area of a triangle is (base * height) / 2. In our case, the base is one side, which we'll call "Moe," and the height is the height of the pyramid.
Since we already know the total area of the four triangular faces is 80 in², we can assume each triangular face has an area of 20 in².
Now, let's solve the mystery of Moe's length. To find the length of one side, we need to find Moe.
We know that the surface area of the square pyramid is 116 in². This means the area of the square base (Moe²) plus the combined area of the four triangular faces (4 * 20 in²) should equal 116 in².
Mathematically, we can write the equation as: Moe² + 4 * 20 in² = 116 in².
By simplifying the equation, we get: Moe² + 80 in² = 116 in².
Subtracting 80 in² from both sides gives us: Moe² = 36 in².
Now, we're just a step away from revealing Moe's true length. By taking the square root of both sides, we find that Moe = 6 in!
So, the comical side length of our square pyramid is 6 in! Larry, Curly, and Moe have solved the mystery together.
To solve this problem, we can use the formula for the surface area of a square pyramid, which is given by:
Surface Area = Base Area + Lateral Area
Since the pyramid is a square pyramid, the base area is equal to the length of one side squared. Let's call that side length "s".
Given: Surface Area = 116 in^2
Lateral Area = 80 in^2
We can substitute these values into the formula:
116 = s^2 + 4(80)
Now, let's simplify the equation and solve for "s":
116 = s^2 + 320
Rearranging the equation:
s^2 = 116 - 320
s^2 = -204
Since the value of s squared is negative, this means that the length of one of the sides cannot be a real number. Therefore, there is no real solution to this problem.
To solve this problem, we need to use the formula for the surface area of a square pyramid:
Surface area = Base area + 4 * triangular face area
We are given that the surface area of the square pyramid is 116 in² and the total area of the four triangular faces is 80 in².
Let's start by finding the base area. Since the base of a square pyramid is in the shape of a square, the base area can be calculated as the length of one of the sides squared.
Let's assume that the length of one of the sides of the square pyramid is 's'. Therefore, the base area is s².
Now, we can substitute these values into the surface area formula:
116 = s² + 4 * 80
Let's simplify the equation:
116 = s² + 320
Subtract 320 from both sides:
s² = 116 - 320
s² = -204
We can see that we obtained a negative value for s², which is not possible for a positive length. Therefore, there is no solution to this problem.
Please note that there might be an error in the given problem, and I would recommend verifying the values provided.