Lisa is having guests over for a game night. She has two tables that are in the shape of rhombuses. The two tables are the same size. She wants to know how many guests she can seat if she pushes her tables together.
The length of one edge of a table is 6 feet. Each person needs 2 feet of space. No person is seated in the corner of a table.
In the picture below, the length of one edge is highlighted and labeled.
I'm sorry, but I cannot see the picture you are referring to. However, I can still help you determine the number of guests Lisa can seat based on the information provided.
Since each person needs 2 feet of space and the length of one edge of the table is 6 feet, 3 people can be seated on each side of a single rhombus table (6 feet / 2 feet per person = 3 people per side). If Lisa pushes the two tables together, it forms an 8-sided shape (octagon), and since no person is seated in the corner of a table, she can seat 3 people on each of the 6 non-corner sides.
So Lisa can seat 18 guests in total (3 people per side × 6 sides = 18 people).
To find out how many guests Lisa can seat when she pushes her tables together, we first need to determine the area of each table.
Since the shape of the tables is a rhombus, the area can be calculated using the formula:
Area = (diagonal1 * diagonal2) / 2
However, in this case, we are given the length of one edge. To find the lengths of the diagonals, we can use the Pythagorean theorem. In a rhombus, the diagonals are perpendicular bisectors of each other.
Let's calculate the lengths of the diagonals:
Using the given table edge length of 6 feet, we can calculate the length of the diagonals (d) using the Pythagorean theorem.
d^2 = 6^2 + 6^2 (both diagonals are equal)
d^2 = 36 + 36
d^2 = 72
Taking the square root of both sides:
d ≈ 8.4853 feet (rounded to four decimal places)
Now that we know the length of each diagonal, we can find the area of one table:
Area = (diagonal1 * diagonal2) / 2
Area = (8.4853 * 8.4853) / 2
Area ≈ 36.0006 square feet (rounded to four decimal places)
Since both tables are the same size, we can double the area to find the total space available:
Total Area = 36.0006 * 2
Total Area ≈ 72.0012 square feet (rounded to four decimal places)
Now we need to calculate the number of guests Lisa can seat. Each person needs 2 feet of space, so we divide the total area by 2:
Number of Guests = Total Area / 2
Number of Guests ≈ 72.0012 / 2
Number of Guests ≈ 36.0006 (rounded to four decimal places)
Therefore, Lisa can seat approximately 36 guests when she pushes her tables together.
To find out how many guests Lisa can seat when she pushes her tables together, we first need to determine the area available for seating.
A rhombus is a four-sided polygon with equal-length sides. In this case, each side of the rhombus-shaped table is 6 feet long.
To calculate the area of a rhombus, we need to know the length of the diagonals. However, the diagonal length is not given in the question. So, we would need to use the side length and some basic geometry principles to find the diagonal length.
In a rhombus, opposite angles are congruent, and the diagonals bisect these angles. Since we know the side length, we can divide the rhombus into two congruent triangles by drawing one of the diagonals.
To find the length of the diagonal, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).
In this case, the diagonal of the rhombus acts as the hypotenuse of the right triangle formed by the side length and half the diagonal length. The length of the side is given as 6 feet.
Let's call the length of half the diagonal x. So, we have the following equation:
(2x)^2 = 6^2 + x^2
Simplifying the equation:
4x^2 = 36 + x^2
Combining like terms:
3x^2 = 36
Dividing both sides by 3:
x^2 = 12
Taking the square root of both sides:
x = √12
x = 2√3
Now that we know the length of half the diagonal, we can find the entire diagonal by doubling it:
Diagonal length = 2 * 2√3
= 4√3
The area of a rhombus can be calculated by multiplying the lengths of its diagonals and dividing by 2:
Area = (Diagonal1 * Diagonal2) / 2
In this case, Diagonal1 and Diagonal2 are the same length, which is 4√3. So, we can calculate the area:
Area = (4√3 * 4√3) / 2
= (16 * 3) / 2
= 48 / 2
= 24 square feet
Now that we know the area available for seating is 24 square feet, we can calculate the number of guests Lisa can seat.
Each person needs 2 feet of space, so we divide the available area by 2:
Number of guests = Area / Space per person
= 24 / 2
= 12 guests
Therefore, Lisa can seat 12 guests when she pushes her tables together.