### Video Transcript

If 𝑏 is the middle proportion between 𝑎 and 𝑐, then which of the following is equal to 𝑏 cubed minus 49𝑐 cubed over 𝑎 cubed minus 49𝑏 cubed? Is it (A) 𝑏 squared over 𝑐 squared, (B) 𝑐 squared over 𝑏 squared, option (C) 𝑏 cubed over 𝑐 cubed, or (D) 𝑐 cubed over 𝑏 cubed?

We begin by recalling our definition of the middle proportion. We know that if 𝑏 is the middle proportion between 𝑎 and 𝑐, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐. In this question, we need to decide which of the four options are equal to 𝑏 cubed minus 49𝑐 cubed over 𝑎 cubed minus 49𝑏 cubed. We notice that none of the four options contain the letter 𝑎. Multiplying both sides of the equation, 𝑎 over 𝑏 equals 𝑏 over 𝑐 gives us 𝑎 is equal to 𝑏 squared over 𝑐. Cubing both sides of this equation, we have 𝑎 cubed is equal to 𝑏 squared over 𝑐 all cubed.

The right-hand side is the same as 𝑏 squared over 𝑐 multiplied by 𝑏 squared over 𝑐 multiplied by 𝑏 squared over 𝑐. Using our laws of indices or exponents, this simplifies to 𝑏 to the sixth power over 𝑐 cubed. We can now substitute our expression for 𝑎 cubed into the expression in the question. This gives us 𝑏 cubed minus 49𝑐 cubed over 𝑏 to the sixth power over 𝑐 cubed minus 49𝑏 cubed. One way to simplify this expression in order to eliminate the fraction on the denominator is to multiply each of the four terms by 𝑐 cubed. On the numerator, we have 𝑏 cubed 𝑐 cubed minus 49 multiplied by 𝑐 to the sixth power. The denominator becomes 𝑏 to the sixth power minus 49𝑏 cubed 𝑐 cubed.

We can now take out a common factor of 𝑐 cubed from the numerator and a common factor of 𝑏 cubed from the denominator. The numerator becomes 𝑐 cubed multiplied by 𝑏 cubed minus 49𝑐 cubed. And the denominator becomes 𝑏 cubed multiplied by 𝑏 cubed minus 49𝑐 cubed. We notice that the expression inside the parentheses is the same on the numerator and denominator. Canceling this leaves us with 𝑐 cubed over 𝑏 cubed.

We can therefore conclude that if 𝑏 is the middle proportion between 𝑎 and 𝑐, then the expression 𝑏 cubed minus 49𝑐 cubed over 𝑎 cubed minus 49𝑏 cubed is equal to 𝑐 cubed over 𝑏 cubed. The correct answer is option (D).