Landscape is designing a display of flowers for an area in a public park. The flower seeds will be planted at points lie on a circle that has a diameter of 8 feet. The point where seed is planted must be at least 2 feet away from seeds on the either side of it.
b. After planting the flower seeds, the landscaper has 20 seed left over. The landscaper wants to plant all, the remaining seeds in another circle so that the seeds are 2 feet apart. What is the diameter of the smallest circle that the landscaper Canute to plant all of the remaining seeds.
I am struck in this for an hour but I did not get. I need help.
the circumference is 8π ft. 2 ft per flower means you get 8π/2 = 4π ≈ 12 plants
20 seeds @ 2 ft/seed means a circle with a 40 ft circumference. So, the diameter is 40/π ≈ 12.75 ft
thank you
Well, it seems like you're in a bit of a "bot-anical" conundrum! Let's see if I can help sprout some answers for you.
For the first part of your question, we know that the diameter of the initial circle is 8 feet. But since each seed must be at least 2 feet away from the ones on either side, we need to find out how many seeds can fit on this circle.
If we imagine placing the first seed at the top of the circle, we can divide the circle into smaller arcs, each measuring 2 feet. By doing some quick math, we can calculate that there would be a total of 8/2 = 4 of these arcs in the entire circle.
Therefore, the landscaper can plant a maximum of 4 seeds on the first circle.
Now, for the second part of your question, we have 20 remaining seeds that need to be planted in another circle. Since each seed needs to be 2 feet apart, we can use a similar approach to determine the diameter of the second circle.
If we divide the circle into 2-foot arcs again, we can see that each arc can accommodate one seed. Therefore, we need a circle with 20 arcs to plant all the remaining seeds.
To find the diameter, we can use the formula: diameter = (number of arcs) * (width of the arcs).
In this case, the number of arcs is 20, and the width of each arc is 2 feet. Plugging those numbers in, we get: diameter = 20 * 2 = 40 feet.
So, the landscaper will need a circle with a diameter of 40 feet to plant all of the remaining seeds.
I hope that "bot-anical" explanation helped! Keep blooming and don't get too "seedy" with all these calculations!
To find the diameter of the smallest circle that the landscaper can use to plant all the remaining seeds, we need to determine the number of seeds needed for this new circle.
Let's start by calculating the circumference of the original circle. Since we know the diameter is 8 feet, the radius (r) would be half of the diameter, which is 8/2 = 4 feet.
The formula for the circumference of a circle is C = 2πr, where π is approximately 3.14. Plugging in the values, we get C = 2 * 3.14 * 4 = 25.12 feet.
Since the point where a seed is planted must be at least 2 feet away from seeds on either side of it, we can calculate the number of seeds on the original circle. The distance between two consecutive seeds is the same as the circumference divided by the required spacing, which is 25.12 / 2 = 12.56 seeds.
Since we cannot have a fraction of a seed, let's round up to the nearest integer to be safe. So, there are 13 seeds on the original circle.
Now, the landscaper has 20 remaining seeds to plant in another circle with a spacing of 2 feet.
Using the same logic as before, we can calculate the number of seeds required on this new circle. The circumference of the new circle would be the number of seeds multiplied by the required spacing, which is 20 * 2 = 40 feet.
The formula for the circumference of a circle is C = 2πr, and we need to find the radius (r) that corresponds to a circumference of 40 feet.
Dividing both sides of the equation by 2π, we get r = C / (2π) = 40 / (2 * 3.14) = 6.37 feet (rounded to two decimal places).
Since the diameter is double the radius, the diameter of the new circle would be 6.37 * 2 = 12.74 feet.
Therefore, the landscaper can use a circle with a diameter of approximately 12.74 feet to plant all the remaining seeds with a spacing of 2 feet.
To solve this problem, let's break it down step by step:
1. Find the number of flower seeds that can be planted in the first circle:
- The diameter of the circle is given as 8 feet, so the radius is half of that, which is 4 feet.
- To find the number of seeds that can be planted, we need to determine the circumference of the circle.
- The formula for the circumference of a circle is C = 2πr, where π is approximately 3.14.
- So, the circumference of the circle is C = 2 * 3.14 * 4 = 25.12 feet.
- Since each seed must be planted 2 feet apart, we can divide the circumference by 2 to find the number of seeds: 25.12 / 2 = 12.56 seeds.
- However, we cannot plant a fraction of a seed, so we'll consider only whole numbers. In this case, we can plant 12 seeds in the first circle.
2. Calculate the circumference of the second circle:
- We know that the landscaper has 20 leftover seeds to plant.
- Each seed must be planted 2 feet apart, so the distance between seeds is 2 feet.
- To find the circumference of the second circle, we need to determine how many seeds can fit on it.
- We can use the formula C = 2πr, where C is the circumference, π is approximately 3.14, and r is the radius.
- Let's assume the radius of the second circle is r feet.
- The number of seeds can be calculated as the circumference of the circle divided by the distance between seeds: C / 2 = πr / 2.
- The number of seeds is also given as 20: 20 = πr / 2.
- To solve for r, we can rearrange the equation: r = (20 * 2) / π.
- Evaluating this, we get r = 40 / 3.14 ≈ 12.74 feet.
3. Determine the diameter of the second circle:
- The diameter of a circle is twice the radius.
- So, the diameter of the second circle is 2 * 12.74 = 25.48 feet.
Therefore, to plant all the remaining seeds with a distance of 2 feet between them, the landscaper needs a circle with a diameter of approximately 25.48 feet.