5).Hat size are determined by the measuring the circumference of the head in either inches or centimeters. Use ratio and proportion to complete the following table.
Hat size Head circumference Head Circumference
7 ½ to nearest 1/5 inch to the nearest centimeter
7 3/8 23 3/5 60
? ?
6). A sample of 144 firecrackers contained 8 duds. How many duds would you expect in a sample of 1584 firecrackers?
7). Find the variation constant and an equation of variation where y varies directly as x and y = 30 when x = 3. The variation constant is k=?
8). The number N of aluminum can used each year varies directly as the number of people P using the cans. If 57 people use 20.919 can s in one year, how many cans are used in a city which contains a population of 1.777.000?
5) To complete the table using ratio and proportion, let's start with the known values:
- Hat size 7 ½ has a head circumference of 23 3/5 inches.
- Head circumference is measured to the nearest 1/5 inch.
To find the head circumference for hat size 7 3/8, we can set up a proportion using cross-multiplication:
(Head Circumference for Hat size 7 ½) / (Hat size 7 ½) = (Head Circumference for Hat size 7 3/8) / (Hat size 7 3/8)
Plugging in the known values:
23 3/5 / 7 ½ = x / 7 3/8
Now, let's solve for x:
Multiply both sides of the proportion by the denominators:
(23 3/5)(7 3/8) = (7 ½)(x)
Convert the mixed numbers to improper fractions:
(118/5)(59/8) = (15/2)(x)
Multiply the fractions:
(118 * 59) / (5 * 8) = (15/2)(x)
Calculate the left side:
6958 / 40 = (15/2)(x)
Simplify the fraction:
173.95 = 7.5x
Divide both sides by 7.5 to solve for x:
x = 173.95 / 7.5
x ≈ 23.19
Therefore, the head circumference for hat size 7 3/8 is approximately 23.19 inches.
To find the head circumference to the nearest centimeter, we can use the conversion factor that 1 inch is approximately equal to 2.54 centimeters:
Head Circumference (in cm) = Head Circumference (in inches) * 2.54
For the given head circumference of 23.19 inches:
Approximate Head Circumference (in cm) = 23.19 * 2.54
Approximate Head Circumference (in cm) ≈ 58.93 cm
Therefore, the head circumference to the nearest centimeter for hat size 7 3/8 is approximately 58.93 cm.
6) To find the number of expected duds in a sample of 1584 firecrackers, we can set up a proportion based on the given information:
(No. of duds in the sample of 144 firecrackers) / (144) = (No. of duds in the sample of 1584 firecrackers) / (1584)
Plugging in the known values:
8 / 144 = x / 1584
Now, let's solve for x:
Multiply both sides of the proportion by the denominators:
144x = 8 * 1584
Simplify the right side:
144x = 12672
Divide both sides by 144 to solve for x:
x = 12672 / 144
x = 88
Therefore, you would expect approximately 88 duds in a sample of 1584 firecrackers.
7) When y varies directly as x, we can represent it as:
y = kx
We are given that when x = 3, y = 30. Plugging these values into the equation, we get:
30 = k * 3
Now, let's solve for k:
Divide both sides by 3:
10 = k
Therefore, the variation constant (k) is 10.
The equation of variation is:
y = 10x
8) When the number of aluminum cans (N) varies directly as the number of people using the cans (P), we can represent it as:
N = kP
We are given that when P = 57, N = 20.919. Plugging these values into the equation, we get:
20.919 = k * 57
Now, let's solve for k:
Divide both sides by 57:
k ≈ 0.3669
Therefore, the variation constant (k) is approximately 0.3669.
To find the number of cans used in a city with a population of 1,777,000, we can use the equation:
N = kP
Plugging in the known values:
N = 0.3669 * 1,777,000
N ≈ 651,428.13
Therefore, approximately 651,428 cans are used in a city with a population of 1,777,000.