A jewelry box is designed such that its length is twice its width and its depth is 2 inches less than its width. The volume of the box is 64 cubic inches.
Use synthetic division to find the roots of the polynomial equation. Are the roots all rational numbers?
Ans: Yes, the only root 4 is rational.
What are the dimensions of the box?
Ans: width is 4, depth is 2 and length is 8.
width=x
length = 2x (2 times the width)
depth = x-2
volume = l*w*d = 64 in^3
x*2x*(x-2)=64
2x^2*(x-2)=64
2x^3-4x^2-64=0
I got x = 4 after some guessing
check: http://www.wolframalpha.com/input/?i=2x%5E3-4x%5E2-64+%3D+0
I agree with you answered for 1 and 2nd part.
To find the dimensions of the jewelry box, we first need to establish the relationships between its length, width, and depth using the given information.
Let's assume the width of the box is "x" inches.
According to the problem, the length of the box is twice its width, which means the length is 2x inches.
The depth of the box is 2 inches less than its width, so the depth can be expressed as (x - 2) inches.
To find the volume of the box, we use the formula for the volume of a rectangular prism: V = length * width * depth.
Substituting the given information into the equation, we get:
64 = (2x) * x * (x - 2)
Simplifying the equation:
64 = 2x^2 * (x - 2)
Divide both sides of the equation by 2:
32 = x^2 * (x - 2)
Now, let's use synthetic division to find the roots of the polynomial equation:
The coefficients of the polynomial equation are 1, 0, -2, and 32.
Set up the synthetic division table, with the coefficients and a guess for a possible root. Let's use x = 4 as our guess:
4 | 1 0 -2 32
| 4 16 56
____________________
1 4 14 88
The remainder is 88, which means 4 is not a root of the equation.
To find the actual root, we can use the Rational Root Theorem, which states that if a polynomial equation with integer coefficients has a rational root, then that root will be of the form p/q, where p is a factor of the constant term (in this case 88) and q is a factor of the leading coefficient (in this case 1).
For the constant term 88, the factors are ±1, ±2, ±4, ±8, ±11, ±22, ±44, and ±88.
For the leading coefficient 1, the factors are ±1.
Therefore, the possible rational roots are ±1, ±2, ±4, ±8, ±11, ±22, ±44, and ±88.
We can try these possible roots using synthetic division until we find one that results in a remainder of 0.
Let's try x = 2 as a possible root:
2 | 1 0 -2 32
| 2 4 4
______________________
1 2 2 36
The remainder is 36, which means 2 is not a root of the equation.
Continuing to try the other possible rational roots, we find that none of them result in a remainder of 0. Therefore, the polynomial equation does not have any rational roots other than 4.
Now, we know that the width of the box is 4 inches, and we can substitute this value back into the expressions for the length and depth:
Length = 2 * width = 2 * 4 = 8
Depth = width - 2 = 4 - 2 = 2
So, the dimensions of the jewelry box are:
Width = 4 inches
Length = 8 inches
Depth = 2 inches
To find the dimensions of the jewelry box, let's assign variables to the dimensions:
Let the width of the box be x inches.
According to the given information, the length is twice the width, so the length is 2x inches.
The depth is 2 inches less than the width, so the depth is (x - 2) inches.
The volume of the box is given as 64 cubic inches, so we can set up the following equation:
Volume = Length * Width * Depth
64 = 2x * x * (x - 2)
64 = 2x^2 * (x - 2)
Now, let's solve this equation step-by-step.
1. Rearrange the equation:
2x^2 * (x - 2) = 64
2. Divide both sides of the equation by 2:
x^2 * (x - 2) = 32
3. Expand the left side of the equation:
x^3 - 2x^2 = 32
4. Rewrite the equation as a polynomial:
x^3 - 2x^2 - 32 = 0
Now, we can use synthetic division to find the roots of the polynomial equation.
The synthetic division table should look like this:
2 | 1 -2 0 -32
Divide the first coefficient (1) by 2 to get 0.5. Write it above the line.
0.5
Multiply 0.5 by 2 to get 1, then add it to the second coefficient (-2). Write the result (-1) under the line.
0.5
_______________
1 -2 0 -32
-1
Multiply 0.5 by -1 to get -0.5, then add it to the third coefficient (0). Write the result (-0.5) under the line.
0.5
_______________
1 -2 0 -32
-1 -0.5
Multiply 0.5 by -0.5 to get -0.25, then add it to the fourth coefficient (-32). Write the result (-32.25) under the line.
0.5
_______________
1 -2 0 -32
-1 -0.5 -32.25
Since the remainder (-32.25) is not equal to zero, 2 is not a root of the polynomial equation.
Therefore, the only rational root of the polynomial equation is x = 4.
Now, let's substitute x = 4 into the dimensions of the jewelry box:
Width = x = 4 inches
Length = 2x = 2 * 4 = 8 inches
Depth = x - 2 = 4 - 2 = 2 inches
So the dimensions of the box are: width = 4 inches, length = 8 inches, and depth = 2 inches.