Final answer:
To determine the value of x for the volume of a rectangular prism, we used the volume formula and set up a quadratic equation. After simplifying and factoring, we found two potential values for x. Rejecting the negative value, we determined that x = 6.
Step-by-step explanation:
To find the value of x in the dimensions of a rectangular prism with a given volume of 1080 cubic cm, we use the formula for the volume of a rectangular prism, which is length x width x height. Plugging in the given dimensions, we have:
5 cm × (x + 12) cm × (x + 6) cm = 1080 cm³
First, multiply the known dimension:
5(x + 12)(x + 6) = 1080
Since 5 is a factor of 1080, divide both sides by 5 to simplify:
(x + 12)(x + 6) = 216
Now, expand the quadratic equation:
x² + 6x + 12x + 72 = 216
Combine like terms:
x² + 18x + 72 = 216
Subtract 216 from both sides to set the quadratic equal to zero:
x² + 18x - 144 = 0
Using the quadratic formula, or factoring if possible, we can solve for x. Factoring it, we get:
(x + 24)(x - 6) = 0
Setting each factor equal to zero gives us two potential solutions:
x + 24 = 0 or x - 6 = 0
Hence, x = -24 or x = 6
Since a dimension cannot be negative, we disregard x = -24. Therefore, the value of x is:
x = 6