The intersection of the three equal squares is a regular triangle.
(The entire shape has 3 axes of symmetry.) Each square has an area of 8. The triangle has an area of 1. The area of the entire shape is 16. What is the area of the black pentagon?
Let's denote the side length of each square as x. Since the area of a square is equal to the side length squared, we can write:
Area of each square = x^2 = 8
Taking the square root of both sides, we get:
x = √8 = 2√2
Since the intersection of the three equal squares is a regular triangle, we can calculate its area using the formula:
Area of a regular triangle = (s^2√3)/4
where s is the length of each side of the triangle. In this case, s is the side length of each square, so s = 2√2.
Area of the triangle = (2√2^2√3)/4 = 4√3
Given that the area of the entire shape is 16, and the area of the triangle is 1, we can calculate the area of the three black pentagons by subtracting the sum of the areas of the squares and the triangle from the total area.
Area of the three black pentagons = 16 - (3 * 8 + 1) = 16 - (24 + 1) = 16 - 25 = -9
The area of the black pentagon is -9. However, since area cannot be negative, there might be an error in the given information or calculations. Please double-check your question or provide additional information.
Please help in under 10 mins. Directions: Use the given rules to complete the table for each sequence and create the ordered pairs. Plot the ordered pairs on the coordinate plane and connect them in order. Then, write a short description of how the corresponding terms are related.
I apologize for the confusion, but I am unable to generate a table, plot coordinates, or connect them within the given timeframe. However, I can still assist you with understanding how the corresponding terms are related based on the given rules if you provide me with the sequence and the rules.
Solve sec(x) cos(3x)=0 from -pi/2,pi/2
To solve the equation sec(x) cos(3x) = 0 from -π/2 to π/2, we can start by considering the two factors:
1. sec(x) = 0
2. cos(3x) = 0
1. For sec(x) = 0, we know that sec(x) = 1/cos(x). Therefore, when sec(x) = 0, cos(x) must be equal to zero. However, we need to find the values of x that satisfy this condition within the given interval.
Looking at the values of cos(x) within the given interval (-π/2 to π/2), we find that cos(x) = 0 for x = -π/2 and x = π/2.
2. For cos(3x) = 0, we need to find the values of x that make cos(3x) equal to zero within the given interval.
We know that cos(3x) = 0 when 3x = π/2 + kπ/2 or 3x = 3π/2 + kπ/2, where k is an integer.
Solving for x in the first equation, we have:
3x = π/2 + kπ/2
x = (π/2 + kπ/2) / 3
Solving for x in the second equation, we have:
3x = 3π/2 + kπ/2
x = (3π/2 + kπ/2) / 3
We need to find these values of x within the given interval. Plugging in k = 0, 1, -1, 2, -2, we get the following values:
For k = 0:
x = (π/2) / 3 = π/6
For k = 1:
x = (π/2 + π/2) / 3 = π/3
For k = -1:
x = (π/2 - π/2) / 3 = 0
For k = 2:
x = (π/2 + 2π/2) / 3 = 5π/6
For k = -2:
x = (π/2 - 2π/2) / 3 = -π/6
Therefore, the solutions for cos(3x) = 0 within the interval -π/2 to π/2 are x = 0, π/6, π/3, 5π/6, and -π/6.
Now, we need to consider the solutions for sec(x) = 0 and cos(3x) = 0 together.
The solutions that satisfy both conditions are x = -π/2, π/2, and π/6.
Hence, the solutions to the equation sec(x) cos(3x) = 0 from -π/2 to π/2 are x = -π/2, π/6, π/2.
To find the area of the black pentagon, we need to determine the area of each individual square first.
Given that each square has an area of 8, the length of each side can be found by taking the square root of 8:
Side length of the squares = √8 = 2√2
Since the entire shape is composed of three equal squares, the length of each side of the equilateral triangle formed by their intersection can be found by multiplying the side length of the squares by 2:
Side length of the triangle = 2 × 2√2 = 4√2
Using the formula for the area of an equilateral triangle, the area of the triangle can be calculated as:
Area of the triangle = (sqrt(3) / 4) × (side length of the triangle)^2
= (sqrt(3) / 4) × (4√2)^2
= (sqrt(3) / 4) × 16
= 4√3
Since the area of the entire shape is 16 and the area of the triangle is 4√3, the combined area of the three squares without the triangle portion can be determined by subtracting the area of the triangle from the total area:
Combined area of the squares = Total area - Area of the triangle
= 16 - 4√3
= 16 - 4√3
Now, to find the area of the black pentagon, we need to subtract the sum of the areas of the three squares from the combined area of the squares:
Area of the black pentagon = Combined area of the squares - Sum of the areas of the three squares
Since each square has an area of 8, the sum of their areas is:
Sum of the areas of the three squares = 3 × Area of each square
= 3 × 8
= 24
Therefore, the area of the black pentagon is:
Area of the black pentagon = Area of the squares - Sum of square areas
= (16 - 4√3) - 24
= -8 - 4√3
The area of the black pentagon is -8 - 4√3.
To find the area of the black pentagon, we first need to determine the remaining area of the entire shape after subtracting the area of the triangle and the three squares.
Given that each square has an area of 8, the total area of the three squares is 8 + 8 + 8 = 24.
The area of the triangle is given as 1, so we subtract this from the total area of the squares: 24 - 1 = 23.
Now, we need to calculate the area of the black pentagon.
Since the entire shape has 3 axes of symmetry, we can deduce that three congruent triangles make up the black pentagon.
Let's consider one of these congruent triangles. Notice that each triangle shares a side with one of the squares. Also, the height of each triangle is equal to the side length of the square, which is √8.
So, the area of one congruent triangle can be calculated by multiplying its base (which is the side length of the square) by its height: ( √8 ) * ( √8 ) / 2 = 8 / 2 = 4.
Since there are three congruent triangles forming the black pentagon, the total area of the pentagon is 3 * 4 = 12.
Therefore, the area of the black pentagon is 12.