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.