Synthesis: Find a polynomial for the shaded area of each figure. 21t+8 is on top of the outer box and 4t is on the left side of the outer box. On the inter box there is 3t-4 and 2t on the left side of the inter box. the area between the outer box and the inter box is shaded. I am having trouble with this problem and can't quite get it right.
(21t+8)(4t) - (3t-4)(2t) = 78t^2 + 40t
To find the polynomial for the shaded area between the outer and inner boxes, we need to determine the dimensions of the shaded region and then calculate its area.
Let's break down the problem step by step:
1. First, let's identify the dimensions of the outer box. We know that the expression "21t + 8" represents the length of the top side of the outer box, and the expression "4t" represents the length of the left side of the outer box.
2. Now, let's identify the dimensions of the inner box. The expression "3t - 4" represents the length of the top side of the inner box, and the expression "2t" represents the length of the left side of the inner box.
3. To find the length of the shaded region on the top side of the outer box, we subtract the length of the inner box (3t - 4) from the length of the outer box (21t + 8). The resulting expression would be: (21t + 8) - (3t - 4).
4. Similarly, to find the length of the shaded region on the left side of the outer box, we subtract the length of the inner box (2t) from the length of the outer box (4t). The resulting expression would be: (4t) - (2t).
5. Now, we have the dimensions of the shaded region: (21t + 8) - (3t - 4) and (4t) - (2t). To find the area of the shaded region, we multiply these two dimensions together.
6. Expanded, our polynomial for the shaded area would be: (21t + 8 - 3t + 4)(4t - 2t).
7. Simplify the expression: (18t + 12)(2t).
8. Multiply the terms within the parentheses: 36t^2 + 24t.
Therefore, the polynomial for the shaded area between the outer box and the inner box is 36t^2 + 24t.