A box that holds an expensive pen has square ends, and its length is 13 cm longer than its width. The volume of the box is 60 cm^3. Determine the dimensions of the box.
60 = w^2 (w+13)
w^3 + 13 w^2 + 0 w - 60 = 0
try w = 2
I got that from
http://www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php
the other two roots do not work for you (negative)
Ah, the case of the posh pen and the peculiar box! Let's put on our mathematical clown hats and solve this riddle, shall we?
Let's call the width of the box "x" cm. According to the problem, the length of the box is 13 cm longer than the width, so the length would be "x + 13" cm.
Now, we need to find the volume of the box. The formula for the volume of a rectangular prism (which is what this box is) is length times width times height. But in this case, we are told that the box has square ends, so the height will be the same as the width.
Since the volume is given as 60 cm^3, we can write the equation:
(x + 13) * x * x = 60
Simplifying, we get:
x^3 + 13x^2 - 60 = 0
Now, we can put on our mathematical clown shoes and solve this cubic equation! However, the calculations might be a bit too complex for me to handle dynamically. Nevertheless, the dimensions of the box can be found using algebraic techniques or numerical methods.
Good luck, my fellow puzzler! Remember, solving math problems is like juggling; it takes a bit of skill, but with practice, you'll soar through it!
To determine the dimensions of the box, we can use the given information and solve step by step.
Let's suppose the width of the box is "x" cm.
According to the problem, the length of the box is 13 cm longer than its width, so the length would be "x + 13" cm.
The height of the box is not given explicitly, but we can find it by using the volume of the box. The volume of a rectangular box is given by the formula:
Volume = length * width * height
In this case, the volume is given as 60 cm^3, so we can substitute the values we have:
60 cm^3 = (x + 13) cm * x cm * height
Now, we need to solve this equation to find the value of the height.
Multiplying the terms on the right side of the equation gives:
60 cm^3 = x(x + 13) cm^2 * height
Expanding the equation gives:
60 cm^3 = x^2 + 13x cm^2 * height
Now, we can rearrange the equation:
x^2 + 13x cm^2 * height = 60 cm^3
Since we are looking for the dimensions of the box, we are interested in finding the values of "x" and "height".
To solve for "x", we can set the equation equal to zero and solve for "x" using factoring, completing the square, or the quadratic formula. In this case, factoring is the simplest method.
x^2 + 13x cm^2 * height - 60 cm^3 = 0
To factor this quadratic equation, we need to find two numbers that multiply to give -60 and add up to 13. In this case, the numbers are 15 and -4:
(x + 15)(x - 4) = 0
Setting each factor equal to zero gives two possible values for "x":
x + 15 = 0 or x - 4 = 0
Solving each equation gives:
x = -15 or x = 4
Since dimensions cannot be negative, we discard the negative value:
x = 4 cm
So, the width of the box is 4 cm.
Now, we can substitute this value into the original equation to find the height of the box:
x^2 + 13x cm^2 * height = 60 cm^3
4^2 + 13(4) cm^2 * height = 60 cm^3
16 + 52 cm^2 * height = 60 cm^3
52 cm^2 * height = 60 cm^3 - 16 cm^3
52 cm^2 * height = 44 cm^3
height = 44 cm^3 / 52 cm^2
height ≈ 0.85 cm
So, the dimensions of the box are approximately:
Width = 4 cm
Length = 4 + 13 cm = 17 cm
Height ≈ 0.85 cm