The side length of a cube is reduced from 25 inches to 15 inches.
b) Nadya says the new surface area of the smaller cube will decrease by a factor of 0.6 but Sharma says it will decrease by a factor of 0.36. Who is correct? Describe the mistake that the incorrect person is making.
To find out who is correct, let's calculate the surface area of each cube.
The surface area of a cube is given by the formula: 6 * (side length)^2
For the original cube with a side length of 25 inches, its surface area is:
6 * (25)^2 = 6 * 625 = 3750 square inches.
For the smaller cube with a side length of 15 inches, its surface area is:
6 * (15)^2 = 6 * 225 = 1350 square inches.
Now let's compare the decrease in surface area.
The decrease in surface area is determined by dividing the new surface area by the original surface area.
For Nadya's claim:
Decrease in surface area = 1350 / 3750 = 0.36.
For Sharma's claim:
Decrease in surface area = 1350 / 3750 = 0.24.
Neither Nadya nor Sharma's claim is correct. The mistake Nadya is making is that she's dividing the new surface area by the original surface area, instead of subtracting the new surface area from the original surface area and then dividing by the original surface area to get the decrease factor. Therefore, Nadya's claim of a decrease by a factor of 0.6 is incorrect.
On the other hand, Sharma's mistake is that he's calculating the decrease factor incorrectly. He is dividing the new surface area by the original surface area, resulting in a decrease factor of 0.36, which is incorrect. The correct decrease factor should be calculated by dividing the new surface area (1350 square inches) by the original surface area (3750 square inches), resulting in a decrease factor of 0.36. Therefore, Sharma's claim of a decrease by a factor of 0.36 is incorrect as well.
In conclusion, neither Nadya nor Sharma is correct in their claims about the decrease in surface area of the smaller cube.
The Eiffel Tower located in Paris, France is 330 m tall. At a novelty store, you can purchase a giant outdoor 3-D puzzle of the Eiffel Tower. The height of the Eiffel Tower puzzle is 1.75 m and the square base that is about 65cm on each side.
a) Determine the linear scale factor, as a fraction, that was used to create the reduced Eiffel Tower.
To determine the linear scale factor, we compare the heights of the real Eiffel Tower and the puzzle Eiffel Tower.
The real Eiffel Tower is 330 m tall.
The puzzle Eiffel Tower is 1.75 m tall.
To find the linear scale factor, we divide the height of the real Eiffel Tower by the height of the puzzle Eiffel Tower:
Linear scale factor = Height of real Eiffel Tower / Height of puzzle Eiffel Tower
= 330 m / 1.75 m
Simplifying this fraction, we get:
Linear scale factor = 330 / 1.75
= 188.5714285714286
Therefore, the linear scale factor used to create the reduced Eiffel Tower puzzle is approximately 188.5714285714286 or approximately 188.5 (rounded to one decimal place).
The Eiffel Tower located in Paris, France is 330 m tall. At a novelty store, you can purchase a giant outdoor 3-D
puzzle of the Eiffel Tower. The height of the Eiffel Tower puzzle is 1.75 m and the square base that is about 65
cm on each side.
b) Determine the side length of the square base of the actual Eiffel Tower to the nearest hundredth of a meter.
To determine the side length of the square base of the actual Eiffel Tower, we need to use the given measurements:
The side length of the puzzle Eiffel Tower's square base is 65 cm.
We know that the actual Eiffel Tower has a height of 330 m. However, we are not given the exact measurements of the sides of its square base.
To find the side length of the square base of the actual Eiffel Tower, we can set up a proportion using the height of the puzzle Eiffel Tower and the side length of its square base:
Height of actual Eiffel Tower / Side length of actual Eiffel Tower's square base = Height of puzzle Eiffel Tower / Side length of puzzle Eiffel Tower's square base
Let's convert the measurements to the same units. We'll convert the height of the puzzle Eiffel Tower to meters and the side length of its square base to meters:
Height of puzzle Eiffel Tower = 1.75 m
Side length of puzzle Eiffel Tower's square base = 65 cm = 0.65 m (since 1 m = 100 cm)
Now we can set up the proportion:
330 m / x = 1.75 m / 0.65 m
Simplifying the proportion:
(330 m * 0.65 m) / 1.75 m = x
215.25 m^2 / 1.75 m = x
x ≈ 123.00 m
Therefore, the side length of the square base of the actual Eiffel Tower is approximately 123.00 meters (rounded to the nearest hundredth of a meter).
The Eiffel Tower located in Paris, France is 330 m tall. At a novelty store, you can purchase a giant outdoor 3-D puzzle of the Eiffel Tower. The height of the Eiffel Tower puzzle is 1.75 m and the square base that is about 65cm on each side.
c) The surface area of the first floor of the actual Eiffel Tower is 4415 m^2. Determine the surface area of the first floor of the 3-D puzzle tower to the nearest hundredth of a square meter.
To determine the surface area of the first floor of the 3-D puzzle tower, we'll use the concept of scaling.
We know that the height of the actual Eiffel Tower is 330 m, while the height of the puzzle Eiffel Tower is 1.75 m. We can use this height ratio to determine the scale factor for the surface area.
Scale factor for the height = Height of actual Eiffel Tower / Height of puzzle Eiffel Tower
= 330 m / 1.75 m
= 188.57
Next, we'll use this scale factor to calculate the surface area of the first floor of the 3-D puzzle tower.
Let's imagine that the surface area of the first floor of the actual Eiffel Tower is A m^2.
Using the scale factor, the surface area of the first floor of the 3-D puzzle tower, A', can be calculated as:
A' = A / (scale factor)^2
Substituting the known values:
A' = 4415 m^2 / (188.57)^2
Calculating this:
A' ≈ 0.12517 m^2
Therefore, the surface area of the first floor of the 3-D puzzle tower is approximately 0.12517 square meters (rounded to the nearest hundredth of a square meter).
The Eiffel Tower located in Paris, France is 330 m tall. At a novelty store, you can purchase a giant outdoor 3-D puzzle of the Eiffel Tower. The height of the Eiffel Tower puzzle is 1.75 m and the square base that is about 65cm on each side.
d) The 3-D puzzle comes disassembled in a box that has a volume of 0.29m^3. Determine the approximate volume of a box that would be needed to contain the actual Eiffel Tower (disassembled) to the nearest m^3.
To determine the approximate volume of a box that would be needed to contain the actual Eiffel Tower (disassembled), we'll use the concept of scaling.
We know that the height of the actual Eiffel Tower is 330 m, while the height of the puzzle Eiffel Tower is 1.75 m. We can use this height ratio to determine the scale factor for the volume.
Scale factor for the height = Height of actual Eiffel Tower / Height of puzzle Eiffel Tower
= 330 m / 1.75 m
= 188.57
Next, we'll use this scale factor to calculate the volume of the box needed for the actual Eiffel Tower (disassembled).
Let's imagine that the volume of the box needed for the actual Eiffel Tower is V m^3.
Using the scale factor, the volume of the box needed for the 3-D puzzle tower, V', can be calculated as:
V' = V / (scale factor)^3
Given that the volume of the puzzle box is 0.29 m^3, we can set up the following equation:
0.29 m^3 = V / (188.57)^3
Solving for V:
V = 0.29 m^3 * (188.57)^3
Calculating this:
V ≈ 1950431.85 m^3
Therefore, the approximate volume of the box needed to contain the actual Eiffel Tower (disassembled) is approximately 1,950,431.85 cubic meters (rounded to the nearest cubic meter).
To determine who is correct, let's first find the surface area of the original cube and the surface area of the smaller cube.
The surface area of a cube is given by the formula: A = 6s^2, where A is the surface area and s is the side length of the cube.
For the original cube with a side length of 25 inches:
A_original = 6 * (25 inches)^2 = 6 * 625 square inches = 3750 square inches
For the smaller cube with a side length of 15 inches:
A_new = 6 * (15 inches)^2 = 6 * 225 square inches = 1350 square inches
Now, let's calculate the ratio of the new surface area to the original surface area:
Ratio = A_new / A_original = 1350 square inches / 3750 square inches = 0.36
So the correct decrease factor of the new surface area is 0.36.
Nadya's statement that the new surface area decreases by a factor of 0.6 is incorrect. The mistake Nadya is making is using the incorrect ratio. It seems that Nadya may have mistakenly calculated the decrease in side length (15 inches / 25 inches = 0.6), and assumed this ratio applied to the surface area as well.
Sharma's statement that the new surface area decreases by a factor of 0.36 is correct, as obtained from the actual surface area calculations.