Answer:
Fernando does not have enough clay to finish the structure since, the expressions for the volume left are not the same.
Step-by-step explanation:
Let w be the length of the longest slab. Since each slab is 2 inches less than the one beneath it, we have the width of the other 3 slabs as w - 2, w - 2 - 2 = w - 4 and w - 4 - 2 = w - 6 respectively.
Since the base of each slab is a square and has thickness 2 inches, the volume of each slab from largest to smallest is thus 2w², 2(w - 2)², 2(w - 4)² and 2(w - 6)² respectively.
We have that we have half of the container of clay left after forming the first two slabs(which are the bottom-most slabs). Let V be the volume of each clay container. Since we have half left, that is V/2, we have used 2V - V/2 = 3V/2 to make the two bottom-most slabs.
So, 3V/2 = 2w² + 2(w - 2)²
= 2w² + 2(w² - 4w + 4)
= 2w² + 2w² - 8w + 8
= 4w² - 8w + 8
= 4(w² - 2w + 2) (1)
Also we want to know if V/2 is equal to the volume of the two top-most slabs.
So, V/2 = 2(w - 4)² + 2(w - 6)²
= 2(w² - 8w + 16) + 2(w² - 12w + 36)
= 2w² - 16w + 32 + 2w² - 24w + 72
= 4w² - 40w + 104
= 4(w² - 10w + 26)
From (1) V/2 = 4(w² - 2w + 2)/3 ≠ 4(w² - 10w + 26)
<u>Since the expressions for the remaining volume are not the same, Fernando does not have enough clay to finish the structure.</u>
