First, let's determine the dimensions of the bath time rubber ducky.
Let x, y, and z represent the dimensions of the bath time rubber ducky.
Given:
Surface Area = 84 cm^2
Volume = 486 cm^3
Surface Area:
2(xy + xz + yz) = 84
Volume:
xyz = 486
Next, we can solve for the dimensions of the bath time rubber ducky by solving the system of equations.
From the surface area equation:
xy + xz + yz = 42
Solving for z in terms of x and y:
z = 42/(xy) - x - y
Substitute this into the volume equation:
x * y * (42/(xy) - x - y) = 486
42 - x^2 - xy - y^2 = 486
x^2 + xy + y^2 = 42
Let's simplify this further:
x^2 + xy + y^2 = 42 -> (x + y)^2 = 42
x + y = sqrt(42)
Now knowing x + y, let's solve for x and y:
If x + y = sqrt(42), and x * y = 18, then x = 6 and y = 3 (or vice versa)
Now, we know the dimensions of the bath time rubber ducky are 6 cm, 3 cm, and 9 cm.
To get the dimensions of the pool float ducky, we will multiply all dimensions by 16:
New dimensions of the pool float ducky:
6 * 16 = 96 cm
3 * 16 = 48 cm
9 * 16 = 144 cm
Now, let's find the surface area of the pool float ducky:
New Surface Area:
2(96*48 + 96*144 + 48*144) = 2(576 + 13824 + 6912) = 2(20412) = 40824 cm^2
Next, we find the volume of the pool float ducky:
New Volume:
96 * 48 * 144 = 663552 cm^3
Therefore, the surface area of the pool float rubber ducky will be 40824 cm^2, and the volume will be 663552 cm^3.
This bath time rubber ducky has a surface area of 84 cm^2 and a volume of 486 cm^3. 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? Show all steps.
1 answer