If ______ amounts of hot and cold water are mixed, the final temperature will b the _______ of the two _______ (beginning) temperatures.

equal, mean/average, original

To determine the final temperature when hot and cold water are mixed, we need to consider the quantities (amounts) and the initial temperatures of the hot and cold water.

Let's assume that the amount of hot water is denoted by H (in any appropriate unit, such as liters) and the amount of cold water is denoted by C. We need to find the final temperature, which we'll call T.

To solve this, we can apply the principle of heat conservation, known as the Law of Mixtures. According to this principle, the total heat gained by the cold water must be equal to the total heat lost by the hot water, leading to thermal equilibrium.

The equation we can use is:

(Hot water heat) = (Cold water heat)

The heat gained/lost by a substance can be calculated using its specific heat (denoted by "c") and the change in temperature (denoted by ΔT). The equation is:

Heat = (mass) x (specific heat) x (change in temperature)

Now, let's substitute the values into the equation:

(H x cH x ΔTH) = (C x cC x ΔTC)

Here, ΔTH and ΔTC represent the changes in temperatures for hot and cold water, respectively.

If we assume that the initial temperature of the hot water is TH and the initial temperature of the cold water is TC, then the changes in temperature can be written as:

ΔTH = T - TH
ΔTC = T - TC

Substituting these expressions into the previous equation:

(H x cH x (T - TH)) = (C x cC x (T - TC))

Now we have an equation with all the variables. By rearranging and simplifying it, we can solve for the final temperature, T:

(H x cH x T - H x cH x TH) = (C x cC x T - C x cC x TC)

H x cH x T - C x cC x T = H x cH x TH - C x cC x TC

T(H x cH - C x cC) = H x cH x TH - C x cC x TC

T = (H x cH x TH - C x cC x TC) / (H x cH - C x cC)

Now that we have the equation, we can plug in the appropriate values for H, C, TH, TC, cH, and cC to calculate the final temperature, T.