(a) A stand at a local parade is selling flags for observers to wave. The small flags each cost 2/3 the price of each large flag, and each flag costs a whole number of dollars. If Frida bought 5 flags and spent $24.00, how much do each of the small flags cost in dollars?
(b) The American flag has 13 horizontal stripes. If the flag outside of Victor’s house is 4 feet tall and 6 feet wide, how many inches tall is each of the 13 stripes? Express your answer as a mixed number.
(c) On a bright, sunny day, Janine is standing 6 feet 3 inches away from the base of a 20-foot-tall flagpole. She notices that her shadow just touches the base of the flagpole. If Janine is exactly 5 feet tall, how long is the shadow of the flagpole at this same time?
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(a) Let x be the price of each large flag in dollars. Then the price of each small flag is (2/3)x dollars. Frida bought 5 flags for a total of 5x dollars. We know that 5x = 24 because she spent $24. Solving for x, we get x = 4.80. Therefore, the price of each small flag is (2/3)($4.80) = $3.20.
(b) The flag is 4 feet tall, which is 48 inches. Each stripe takes up 1/13 of the total height, so each stripe is (1/13)(48) = 3.69 inches tall. This can be written as a mixed number as 3 11/13 inches tall.
(c) Janine's height and the distance between her and the flagpole form a right triangle, where the flagpole is the vertical side and Janine's shadow is the horizontal side. The ratio of the lengths of the vertical side to the horizontal side is the same as the ratio of Janine's height to the length of her shadow. In other words:
20 / x = 5 / (x - 6'3" or 75 inches)
Cross-multiplying and solving for x, we get x = 24 feet. Therefore, the length of the flagpole's shadow is 24 - 6'3" or 18'9" or 225 inches.
(a) wrong, since each flag costs a whole number of dollars.
If each flag costs a whole number of dollars, we need to find two whole numbers that satisfy the conditions in the problem. Let x be the price of each large flag in dollars. Then the price of each small flag is (2/3)x dollars. Since each flag costs a whole number of dollars, we know that x must be a multiple of 3. We also know that 5 flags cost $24.
Let's try some values of x that are multiples of 3:
- If x = 3, then the price of each small flag is (2/3)($3) = $2, and the total cost of 5 flags is $13, which is too low.
- If x = 6, then the price of each small flag is (2/3)($6) = $4, and the total cost of 5 flags is $26, which is too high.
- If x = 9, then the price of each small flag is (2/3)($9) = $6, and the total cost of 5 flags is $30, which is too high.
- If x = 12, then the price of each small flag is (2/3)($12) = $8, and the total cost of 5 flags is $40, which is too high.
We can continue trying larger values of x, or we can notice that if x = 15, then the price of each small flag is (2/3)($15) = $10, and the total cost of 5 flags is $50, which is too high. Therefore, the only possibility is that x = 18.
If x = 18, then the price of each small flag is (2/3)($18) = $12, and the total cost of 5 flags is $60, which is too high. Therefore, x cannot be 18.
We can try x = 21, but it's clear that this will be too high since the total cost of 5 flags will be at least $70. Therefore, we conclude that there is no solution to this problem if each flag must cost a whole number of dollars.
(a) To find the cost of each small flag, we need to set up an equation using the given information. Let's assume the cost of each small flag is x dollars.
Since each small flag costs 2/3 the price of each large flag, the cost of each large flag would be (3/2)x dollars.
Now, let's analyze the total cost Frida spent on the flags. She bought 5 flags, which consists of some small flags and some large flags.
The total cost can be calculated by adding the cost of each small flag multiplied by the number of small flags, plus the cost of each large flag multiplied by the number of large flags. In mathematical terms, this can be expressed as:
Total cost = (x dollars) * (number of small flags) + ((3/2)x dollars) * (number of large flags)
Since Frida bought 5 flags in total, we can set up another equation relating the number of small flags and large flags:
(number of small flags) + (number of large flags) = 5
Since we have two equations, we can solve the system of equations to find the values of x (cost of each small flag) and (number of small flags). However, we need one more piece of information to determine a unique solution.
(b) To find the height of each of the 13 stripes on the American flag, we need to divide the total height of the flag by the number of stripes.
Given that the flag outside of Victor's house is 4 feet tall and has 13 stripes, each of the 13 stripes would have a height of 4 feet divided by 13. This results in a decimal, which we can convert to a mixed number.
To convert the decimal to a mixed number, we can divide 4 by 13. The quotient is the whole number part of the mixed number (0) and the remainder is the numerator of the fraction (4). The denominator remains the same (13).
Hence, each of the 13 stripes is approximately 0 and 4/13 feet tall.
(c) To find the length of the shadow of the flagpole, we can use the concept of similar triangles. The length of the shadow is proportional to the height of the flagpole.
Given that Janine is standing 6 feet 3 inches away from the base of the 20-foot-tall flagpole, we have the following measurements:
Janine's height (including her shadow) = 5 feet + length of her shadow
Flagpole's height = 20 feet
Since the triangles formed by Janine, her shadow, and the flagpole are similar, the ratios of corresponding sides are equal.
We can set up a proportion to solve for the length of the shadow:
Janine's height / Janine's distance from the flagpole = Flagpole's height / Length of the shadow
In this case, substituting the given values, we have:
(5 feet + length of her shadow) / 6 feet 3 inches = 20 feet / Length of the shadow
To solve this equation, we need to convert the measurements to a consistent unit. Let's convert the 6 feet 3 inches to inches, which is (6 feet * 12 inches/foot) + 3 inches = 75 inches.
Now, we have:
(5 feet + length of her shadow) / 75 inches = 20 feet / Length of the shadow
We can then cross-multiply and solve for the length of the shadow.