Part B: This bath time rubber ducky has a surface area of 84 cm2 and a volume of 486 cm3. The bath time ducky and the pool float ducky are similar figures. If all of the dimensions of the bath time rubber ducky are multiplied by 16 to get this
pool float rubber ducky, what will the surface area and the volume of the pool float ducky be?
Since the pool float ducky is similar to the bath time ducky, all corresponding sides will be in the same proportion.
Let's call the scale factor between the two ducks "k". In this case, k = 16.
To find the new surface area of the pool float ducky, we need to square the scale factor and multiply by the original surface area of the bath time ducky:
Surface area of pool float ducky = k^2 * surface area of bath time ducky
Surface area of pool float ducky = 16^2 * 84 cm^2
Surface area of pool float ducky = 21,504 cm^2
Therefore, the surface area of the pool float ducky will be 21,504 cm^2.
To find the new volume of the pool float ducky, we need to cube the scale factor and multiply by the original volume of the bath time ducky:
Volume of pool float ducky = k^3 * volume of bath time ducky
Volume of pool float ducky = 16^3 * 486 cm^3
Volume of pool float ducky = 1,228,800 cm^3
Therefore, the volume of the pool float ducky will be 1,228,800 cm^3.
To find the surface area and volume of the pool float rubber ducky, which is similar to the bath time rubber ducky with dimensions multiplied by 16, we can use the concept of similarity.
Let's denote the dimensions of the bath time rubber ducky as follows:
Length = L
Width = W
Height = H
Given that the surface area of the bath time rubber ducky is 84 cm^2, we can write the equation:
2(LW + LH + WH) = 84 (Equation 1)
Also, given that the volume of the bath time rubber ducky is 486 cm^3, we have the equation:
V = LWH = 486 (Equation 2)
Since the pool float rubber ducky is similar to the bath time rubber ducky, the ratio of the corresponding sides of the two ducks will be the same.
So, if we multiply each dimension of the bath time rubber ducky by 16, the dimensions of the pool float rubber ducky will be:
Length of pool float rubber ducky = 16L
Width of pool float rubber ducky = 16W
Height of pool float rubber ducky = 16H
To find the surface area (SA') of the pool float rubber ducky, we can use the formula:
SA' = 2((16L)(16W) + (16L)(16H) + (16W)(16H)) (Note: SA' denotes the surface area of the pool float rubber ducky)
Similarly, to find the volume (V') of the pool float rubber ducky, we can use the formula:
V' = (16L)(16W)(16H) (Note: V' denotes the volume of the pool float rubber ducky)
Now, let's calculate the surface area and volume of the pool float rubber ducky.
First, simplify the expressions inside the formulas:
SA' = 2(256LW + 256LH + 256WH)
= 512 LW + 512 LH + 512 WH
V' = 4096LWH
Therefore, the surface area of the pool float rubber ducky will be 512 times the surface area of the bath time rubber ducky, and the volume of the pool float rubber ducky will be 4096 times the volume of the bath time rubber ducky.
To get the specific values, plug in the given surface area and volume of the bath time rubber ducky into the formulas.
Surface Area of the Pool Float Ducky:
SA' = 512(84) = 43,008 cm^2
Volume of the Pool Float Ducky:
V' = 4096(486) = 1,985,536 cm^3
So, the surface area of the pool float rubber ducky will be 43,008 cm^2, and the volume of the pool float rubber ducky will be 1,985,536 cm^3.
To find the surface area and volume of the pool float ducky, we first need to determine the scale factor by which the dimensions of the bath time rubber ducky are multiplied to get the pool float rubber ducky.
Since the bath time rubber ducky and the pool float rubber ducky are similar figures, their corresponding sides are proportional. This means that their surface areas will be proportional to the square of the scale factor, and their volumes will be proportional to the cube of the scale factor.
Let's calculate the scale factor first:
Scale factor = Dimension of the pool float ducky / Dimension of the bath time ducky
Since all dimensions are multiplied by 16, the scale factor is 16.
Now, we can calculate the surface area and volume of the pool float ducky using the scale factor.
Surface area of the pool float ducky = Surface area of the bath time ducky * (Scale factor)^2
= 84 cm^2 * (16)^2
= 84 cm^2 * 256
= 21504 cm^2
Volume of the pool float ducky = Volume of the bath time ducky * (Scale factor)^3
= 486 cm^3 * (16)^3
= 486 cm^3 * 4096
= 1990656 cm^3
Therefore, the surface area of the pool float ducky will be 21504 cm^2 and the volume will be 1990656 cm^3.