1.The population of Loompaland over an 8-year period is shown below. What is the greatest percent increase in the population from any year to the next during this time? Express your answer to the nearest tenth.
2. The circle below has a 144 degree central angle cut out and removed. The remaining sector of the circle is formed into a cone by connecting points A and B from the original circle. The radius of the original circle is 10 cm. What is the volume of the new cone? Express your answer in terms of π.
3. A circle and an equilateral triangle have the same area. What is the ratio of the length of the radius of the circle to the side length of the triangle? Express your answer as a decimal to the nearest hundredth.
4. Two different points from the 4 by 7 array below are chosen as the endpoints of a line segment. How many different lengths could this segment have?
5. Kris flipped a coin 12 times. What is the probability that the number of times the coin landed on heads or the number of times the coin landed on tails was a prime number? Express your answer as a common fraction.
6. In square ABCD below with side lengths 6 cm, two semicircles are centered on the midpoints of sides AD and BC using segments AD and BC as their diameters. Two more semicircles are centered on sides AB and CD such that these semicircles are each tangent to the larger two semicircles centered on segments AD and BC. The area outside these four semicircles and inside the square is shaded. What is the total area of the shaded region? Use 3.1415 as an estimation for π and express your answer as a decimal to the nearest tenth.
Also, the figures "below" are absent.
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1. To find the greatest percent increase in the population from any year to the next, you need to find the largest percentage change between any two consecutive years in the given data.
Start by calculating the percent change from one year to the next for each pair of consecutive years. For example, to find the percent change from year 1 to year 2, you would calculate:
Percent change = ((Population in year 2 - Population in year 1) / Population in year 1) * 100%
Repeat this calculation for each pair of consecutive years and determine the percentage change that yields the highest value. Round your answer to the nearest tenth.
2. To find the volume of the new cone formed from the sector of the circle, you need to use the formula for the volume of a cone:
Volume = (1/3) * π * radius^2 * height
The radius of the original circle is given as 10 cm. To find the height of the cone, you need to determine the length of the arc that was removed. Since the central angle cut out is 144 degrees, that represents a fraction of the entire circumference of the circle.
Fraction of circumference = 144 degrees / 360 degrees = 2/5
Now, calculate the length of the arc that was removed:
Arc length = Fraction of circumference * circumference
Arc length = (2/5) * (2 * π * radius)
The remaining sector of the circle is then formed into a cone by connecting points A and B. The height of the cone is equal to the radius of the original circle.
With the radius and height of the cone known, you can calculate the volume using the formula mentioned earlier. Express your answer in terms of π.
3. To find the ratio of the length of the radius of the circle to the side length of the equilateral triangle, you need to know the formula for the area of a circle and the area of an equilateral triangle.
For a circle, the area is given by:
Area of circle = π * radius^2
For an equilateral triangle, the area is given by:
Area of equilateral triangle = (sqrt(3) / 4) * side length^2
Since the two areas are equal:
π * radius^2 = (sqrt(3) / 4) * side length^2
You can solve this equation for the ratio of radius to side length.
4. To find the number of different lengths a line segment can have when choosing two different points from the given 4 by 7 array, you need to consider the possible horizontal and vertical distances between the chosen points.
For horizontal distances, there are 3 possible lengths: 1 unit, 2 units, and 3 units.
For vertical distances, there are 6 possible lengths: 1 unit, 2 units, 3 units, 4 units, 5 units, and 6 units.
To find the total number of different lengths, you need to count the combinations of horizontal and vertical distances between the chosen points. Multiply the number of possible horizontal distances by the number of possible vertical distances to get the total number of different lengths.
5. To find the probability that the number of times the coin landed on heads or tails is a prime number when flipping a coin 12 times, you need to count the number of prime numbers within the range of possible outcomes.
The possible outcomes range from 0 to 12, as there are 12 coin flips. Count the number of prime numbers within this range (for example, 2, 3, 5, 7, 11) and divide it by the total number of possible outcomes (13 in this case, including 0).
6. To find the total area of the shaded region in the given diagram, you need to calculate the areas of the various shapes and subtract them accordingly.
Start by calculating the area of the square ABCD using the formula: Area = side length^2.
Next, calculate the areas of the four semicircles using the formula: Area = (1/2) * π * radius^2.
To find the radius of the larger two semicircles on segments AD and BC, divide the side length of the square by 2. To find the radius of the smaller two semicircles on sides AB and CD, divide the side length of the square by 4.
Once you have the areas of the four semicircles, subtract them from the area of the square to find the total area of the shaded region. Use π as an estimation for the value of pi and round your answer to the nearest tenth.