Water reaches its maximum density above its freezing point. The volume V (in cubic diameters) of 1 kg of water at temperature T between 0 degrees C and 30 degrees C can be approximated by the formula V=999.87 - 0.06426T + 0.0085043T^2 - 0.0000679T^3. Find the temperature at which water has its maximum density
To find the temperature at which water has its maximum density according to the given formula, we need to determine the maximum point of the volume function V(T).
The given formula for the volume of water as a function of temperature is:
V = 999.87 - 0.06426T + 0.0085043T^2 - 0.0000679T^3
To find the temperature at which water has its maximum density, we need to find the maximum point of the volume function, which can be done by taking the derivative of the volume function with respect to T and setting it equal to zero.
Step 1: Take the derivative of V(T) with respect to T:
dV/dT = -0.06426 + 0.0170086T - 0.0002037T^2
Step 2: Set the derivative equal to zero and solve for T:
-0.06426 + 0.0170086T - 0.0002037T^2 = 0
Step 3: Simplify the equation by multiplying through by -10000 to eliminate decimals:
642.6 - 170.086T + 2.037T^2 = 0
Step 4: Rearrange the equation in standard quadratic form:
2.037T^2 - 170.086T + 642.6 = 0
Step 5: Solve the quadratic equation for T. This can be done by factoring, completing the square, or using the quadratic formula:
Using the quadratic formula, T = (-b ± √(b^2 - 4ac)) / (2a)
For the quadratic equation 2.037T^2 - 170.086T + 642.6 = 0, a = 2.037, b = -170.086, and c = 642.6.
T = (-(-170.086) ± √((-170.086)^2 - 4 * 2.037 * 642.6)) / (2 * 2.037)
Simplifying this equation will give you two possible values for T. To determine which value corresponds to the temperature at which water has its maximum density, we need to select the temperature within the given range of 0 to 30 degrees Celsius.
After calculating the values, the temperature at which water has its maximum density is the value between 0 and 30 degrees Celsius that solves the equation.