A right rectangular tank with a 12 unit by 8 unit base is filled with water to a depth of 5 units. The water rises 1.5 units when a solid cube is completely submerged in the tank. In simplest radical form, the edge of the cube is k*cube root of w units. Find the value of k plus w.
The volume of the cube raises the water 1.5 units
so the volume of the cube is:
1.5 * 12 * 8 units cubed
so
V = 144 units^3
that is the cube of the edge of the cube
144 = (k* w^1/3)^3 = k^3 w
That is all I can do with the information given.
Oh, I see from your earlier questions that you are doing only integer answers
Well, 144 is a handy number being 12^2
so
2 * 2 * 3 * 2 * 2 *3
two will have to be cubed
8 * 18 is 2^3 * 18
So I think k = 2 and w = 18
Let's break down the problem step by step:
Step 1: Find the volume of the tank.
The volume of a rectangular prism can be calculated by multiplying the length, width, and height. In this case, the base has dimensions of 12 units by 8 units, and the tank is filled to a depth of 5 units. Therefore, the volume is 12 * 8 * 5 = 480 cubic units.
Step 2: Find the volume of the cube.
Since the cube is submerged in the water and causes it to rise 1.5 units, we can assume that the volume of the cube is equal to the volume of water displaced. Given that water has a density of 1 g/cm³ or 1 kg/L, we can say that the volume of the cube is 1.5 * 480 = 720 cubic units.
Step 3: Find the edge length of the cube.
To find the edge length of the cube, we can use the formula for the volume of a cube, which is s³, where s is the length of the edge. So, we have the equation s³ = 720.
Step 4: Simplify the equation.
Taking the cube root of both sides, we get s = ∛720.
To simplify ∛720, we can break it down into prime factors: ∛(2 * 2 * 2 * 2 * 3 * 3 * 5) = 2 * 2 * ∛(2 * 3 * 3 * 5) = 4∛(90).
Step 5: Calculate the value of k + w.
From step 4, we have the edge length of the cube as 4∛(90) units.
Therefore, k + w = 4 + 90 = 94.
So, the value of k + w is 94.
To find the value of k plus w, we need to determine the values of k and w separately.
Let's start by finding k.
The problem states that the edge length of the cube is represented by k times the cube root of w units. Let's assume the edge length of the cube is "x" units.
When the cube is submerged in the tank, the water level rises by 1.5 units. This means the volume of the cube is equal to the volume of the water displaced, which is equal to the increase in the tank's volume.
The increase in volume of the tank can be calculated by multiplying the area of the base (12 units by 8 units) by the rise in water level (1.5 units).
Increase in volume = 12 units * 8 units * 1.5 units = 144 units^3
Since the cube has all sides of equal length, its volume is given by:
Volume of cube = x^3
Setting the volume of the cube equal to the increase in volume of the tank:
x^3 = 144
To find the value of x, we need to find the cube root of 144.
The cube root of 144 can be simplified as follows:
Cube root of 144 = 12 * cube root of 1
= 12
Therefore, the value of k is 12.
Now, let's find the value of w.
Since the edge length of the cube is represented by k times the cube root of w units, we can substitute k = 12 into the equation.
So, the equation becomes:
x = 12 * cube root of w
But we already found that x = 12.
Substituting x = 12 into the equation:
12 = 12 * cube root of w
Dividing both sides by 12:
1 = cube root of w
To find the value of w, we need to cube both sides of the equation.
1^3 = (cube root of w)^3
1 = w
Therefore, the value of w is 1.
Finally, to find the value of k plus w:
k + w = 12 + 1 = 13
So, the value of k plus w is 13.