Question

the diagram of the horse training center shows its measurements in meters. lin and noah each write a different expression to find the area of the training center. lins expression: 40y + 40(20) noahs expression: 40(y + 20) first, describe lins expression. area of yard + area of arena next, describe noahs expression. ? × ?

Explanation:

Step1: Analyze Lin's expression

Lin's expression \(40y + 40(20)\) is based on the sum of the areas of two rectangles. The area of a rectangle is \(length\times width\). For the "yard" part, with length \(40\) and width \(y\), its area is \(40y\). For the "arena" part, with length \(40\) and width \(20\), its area is \(40\times20 = 40(20)\). So Lin's expression is the sum of the area of the yard and the area of the arena.

Step2: Analyze Noah's expression

Noah's expression \(40(y + 20)\) uses the distributive property \(a(b + c)=ab+ac\). Here, if we consider the combined figure as a single rectangle with length \(40\) and width \((y + 20)\) (since the total width of the combined "yard - arena" figure is \(y+20\) and length is \(40\)), by the formula for the area of a rectangle \(A = length\times width\), we get \(A=40\times(y + 20)=40(y + 20)\).

Answer:

Noah's expression represents the area of a single rectangle with length \(40\) and width \((y + 20)\) (where \(y+20\) is the sum of the widths of the "yard" and "arena" parts).