The smallest figurine in a gift shop is 2 inches tall. The height of each figurine is twice the height of the previous figurine. Write a power to represent the height of the tallest figurine. Then find the height.
A square painting measures 2 meters on each side. What is the area
of the painting in square centimeters?
To represent the height of the tallest figurine, we can use the power of 2 since each figurine's height is twice the height of the previous figurine. Let's call the number of figurines in the gift shop n.
The height of the first figurine is 2 inches (2^0 = 1).
The height of the second figurine is twice the height of the previous figurine, which is 2 * 2 = 4 inches (2^1).
The height of the third figurine is twice the height of the previous figurine, which is 2 * 4 = 8 inches (2^2).
Continuing this pattern, we can say that the height of the nth figurine is 2^(n-1) inches.
We need to find the height of the tallest figurine, which means finding the value of n for the tallest figurine.
Since the height of each figurine doubles with each new figurine, we need to find the value of n when the height is the tallest. This occurs when the height of the figurine reaches its maximum limit in the gift shop.
To find the height, we need to solve the equation 2^(n-1) = tallest figurine height.
Without knowing the specific tallest figurine height, we cannot determine the value of n or calculate the height.
To represent the height of the tallest figurine as a power, we can consider the first figurine as the base height and the number of figurines as the exponent.
Let's assume the height of the first figurine is h inches. According to the given information, the height of each figurine is twice the height of the previous figurine. Therefore, the height of the second figurine would be 2h inches, the height of the third figurine would be 2 * 2h = 2^2 * h inches, and so on.
Since the height of each figurine is twice the height of the previous one, we can write the height of the tallest figurine as a power of 2: 2^n * h, where n represents the number of figurines.
Now, let's find the height of the tallest figurine. We know that the height of the smallest figurine is 2 inches, and each figurine's height is twice the height of the previous one.
If we let n represent the number of figurines, we can determine the value of n by finding the relationship between the height of the tallest figurine and the smallest figurine:
2^n * h = height of tallest figurine
In this case, the smallest figurine has a height of 2 inches. Using the given information, we can derive the value of h by rearranging the equation:
2^n * h = 2
h = 2 / (2^n)
Since the height of the smallest figurine is 2 inches, we have:
2 = 2 / (2^n)
Cross-multiplying, we get:
2 * 2^n = 2
Simplifying further:
2^n = 1
To solve for n, we take the logarithm of both sides of the equation with base 2:
log base 2 (2^n) = log base 2 (1)
n = log base 2 (1)
Since any number raised to the power of 0 is 1, thus log base 2 (1) = 0:
n = 0
Therefore, the height of the tallest figurine is 2^0 * h, which simply means the height of the smallest figurine:
Height of the tallest figurine = 2^0 * 2 = 2 inches.