A rectangular cardboard is 8cm long and 6cm wide.what is the least number of beads you can arrange on the board such that they are at least two of the beads that are less than √10cm apart?
Please show step...
One at each of two opposite corners gives sqrt (64+36) = 10 cm apart
so two
Why did the bead go on a vacation?
Because it needed some R&R, of course!
Now, let's get back to your question. To find the least number of beads that are at least √10cm (approximately 3.16cm) apart, we can start by placing one bead at each corner of the rectangular cardboard. So we would have four beads placed.
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Now, let's calculate the distance between two diagonal beads to determine if they are at least √10cm apart. Using the Pythagorean Theorem (a^2 + b^2 = c^2), we can calculate the distance between the two diagonally opposite beads:
c^2 = (8cm)^2 + (6cm)^2
c^2 = 64cm^2 + 36cm^2
c^2 = 100cm^2
To find c, we take the square root of both sides:
c ≈ √100cm^2
c ≈ 10cm
Since the distance between the diagonal beads is exactly 10cm, we need to place another bead at a distance of less than √10cm from the existing ones. We can accomplish this by placing a bead between two diagonal beads, like this:
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Now, the distance between any two of the four beads is less than √10cm. Therefore, the least number of beads we can arrange is 5.
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I hope this answer tickled your funny bone while also providing a helpful solution!
To find the least number of beads that can be arranged on the board such that there are at least two beads that are less than √10 cm apart, we can approach this problem by using a geometric approach.
Step 1: Calculate the distance between any two corners of the rectangle.
- The distance between two corners of a rectangle can be calculated using the Pythagorean theorem.
- Given that the cardboard is 8 cm long and 6 cm wide, we can calculate the distance between two corners as follows:
Distance = √(length^2 + width^2)
= √(8^2 + 6^2)
= √(64 + 36)
= √100
= 10 cm
The distance between any two corners of the rectangle is 10 cm.
Step 2: Determine the maximum possible distance between two beads.
- To ensure that there are at least two beads that are less than √10 cm apart, we need to consider the maximum distance between two beads.
- The maximum possible distance is half the distance between two corners of the rectangle.
Maximum Distance = 1/2 * Distance
= 1/2 * 10 cm
= 5 cm
The maximum possible distance between two beads is 5 cm.
Step 3: Determine the minimum number of beads required.
- We need to find the minimum number of beads required to cover the rectangle such that there are at least two beads that are less than 5 cm apart.
- To do this, we divide the length and width of the rectangle by the maximum possible distance between two beads, and round up to the nearest whole number.
Minimum Number of Beads = ceil(length / Maximum Distance) * ceil(width / Maximum Distance)
= ceil(8 cm / 5 cm) * ceil(6 cm / 5 cm)
= ceil(1.6) * ceil(1.2)
= 2 * 2
= 4
The minimum number of beads required is 4.
Therefore, the least number of beads you can arrange on the board such that there are at least two beads that are less than √10 cm apart is 4.
To determine the least number of beads that can be arranged on the rectangular cardboard such that there are at least two beads less than √10cm apart, we need to consider the maximum possible distance between any two beads.
Step 1: Calculate the maximum possible distance between any two points on the rectangular cardboard.
The diagonal of a rectangle can be calculated using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the length of the rectangle is 8cm, and the width is 6cm. So, using the Pythagorean theorem, we can calculate the diagonal length:
diagonal length = √(length^2 + width^2)
diagonal length = √(8^2 + 6^2)
diagonal length = √(64+36)
diagonal length = √100
diagonal length = 10cm
Step 2: Determine the maximum possible number of beads that can be placed such that no two beads are less than √10cm apart.
Since we want to make sure that there are at least two beads less than √10cm apart, we need to make sure that the distance between all pairs of beads is greater than or equal to √10cm.
The maximum possible distance between any two beads is √10cm, so the minimum number of beads required can be calculated by dividing the diagonal length by √10cm:
minimum number of beads = diagonal length / √10cm
minimum number of beads = 10cm / √10cm
minimum number of beads = 10 / √10
minimum number of beads ≈ 3.16
Since we cannot have a fraction of a bead, we need to round up the number to the nearest whole number:
minimum number of beads = 4
Therefore, the least number of beads that can be arranged on the rectangular cardboard such that there are at least two beads less than √10cm apart is 4.