The length of a child's square-based jewelry box is 5cm more than its height. The box has a capacity of 500cm^3. Solve a polynomial equation to determine the dimensions of the box.
l=w=h+5 or h= l-5
500=Lwh= l^2(l-5)
l^3-L^2-500=0
I would solve that graphically on your calculator, see were L satisfies the function...about 7.6cm
Let's assume the height of the jewelry box is x cm.
According to the given information, the length of the box is 5 cm more than its height, so the length would be x + 5 cm.
Since the jewelry box is square-based, the width would also be x + 5 cm.
The volume of a square-based box is found using the formula V = length * width * height, so we can set up the equation as follows:
(x + 5) * (x + 5) * x = 500
Expanding the equation, we get:
(x^2 + 10x + 25) * x = 500
Multiplying further, we have:
x^3 + 10x^2 + 25x = 500
Rearranging the equation to obtain a polynomial equation in standard form:
x^3 + 10x^2 + 25x - 500 = 0
This is the polynomial equation that we can solve to determine the dimensions of the box.
To solve this problem, we can set up a polynomial equation based on the given information and then solve the equation for the dimensions of the box.
Let's assume that the height of the box is h cm. According to the problem, the length of the box is 5 cm more than the height, so the length can be represented as (h + 5) cm.
The volume of a square-based box can be calculated by multiplying the width (which is the same as the height) by the length and the height. Since the box has a capacity of 500 cm^3, we can set up the equation:
Volume = Height * Length * Height = 500 cm^3
Plugging in the given values, we have:
(h * (h + 5) * h) = 500
Now, let's simplify and solve this equation:
h^3 + 5h^2 - 500 = 0
This is a cubic polynomial equation. To solve it, we can use various methods such as factoring, synthetic division, or the rational root theorem. However, this particular equation is not easy to factor, so let's use another method known as numerical approximation.
Using a graphing calculator or a mathematical software, we can plot the function f(h) = h^3 + 5h^2 - 500 and find its roots. In this case, we find that one of the roots is approximately h ≈ 7.403.
Since the height cannot be negative, we discard any negative solutions. Therefore, the height of the box is approximately 7.403 cm.
Using this height value, we can find the length of the box:
Length = Height + 5 = 7.403 + 5 = 12.403 cm
So, the dimensions of the box are approximately 7.403 cm (height) and 12.403 cm (length).
bob's equation should have been
l^3 - 5l^2 - 500 = 0
(l-10)(l^2 - 5l + 50) = 0
l = 10 , since the quadratic has complex roots
the box is 10 by 10 by 5