Final answer:
Proportionality constants are determined by substituting known values into equations representing direct, joint, and inverse relationships. In this case, a student learns how to calculate these constants for various proportional relationships and also finds two numbers with a given sum and difference.
Step-by-step explanation:
In the study of mathematical relationships, we may come across various types of proportionalities. When we say that t varies jointly as h and m, we write this as t = k × h × m, where k is the proportionality constant. For p varies directly as n, the relationship is given by p = k × n.
If we are given that j is jointly proportional to p and m, and are given values for j, p, and m at a certain point, we can find the constant k by rearranging the equation to j = k × p × m. Substituting the values, we get 6 = k × 5 × 4, which gives us k = 6 / (5 × 4) = 0.3.
For the relationship where q varies directly as c, we follow the equation q = k × c. Given that c = 4 when q = 7, we can solve for the constant k, yielding k = 7 / 4 = 1.75.
In an inverse proportionality, such as z varies inversely as x, the relationship is described by z = k / x. Substituting the known values z = 8 and x = 4, we find k = z × x = 8 × 4 = 32.
To find two numbers a and b whose sum a+b is 4 and whose difference a−b is 2, we solve the following system of equations:
a + b = 4
a - b = 2. Adding these two equations, we get 2a = 6, which gives us a = 3. Substituting a into one of the equations to find b gives us b = 1.