kara is building a sandbox shaped like a kite for her nephew. the top two sides of the sandbox are 29 inches…

Explanation:

Step1: Recall kite - diagonal properties

The diagonals of a kite are perpendicular, and one diagonal bisects the other. Let the intersection of the diagonals (AC) and (DB) be point (O). Let (DB = 40) inches, so (DO=\frac{DB}{2}=20) inches and (OB = 20) inches.

Step2: Use the Pythagorean theorem in right - triangle (AOD)

In right - triangle (AOD), (AD = 29) inches and (DO = 20) inches. By the Pythagorean theorem (a^{2}+b^{2}=c^{2}), where (c = AD), (a = DO), and (b = AO). So (AO=\sqrt{AD^{2}-DO^{2}}). Substitute (AD = 29) and (DO = 20) into the formula: (AO=\sqrt{29^{2}-20^{2}}=\sqrt{(29 + 20)(29 - 20)}=\sqrt{49\times9}=\sqrt{441}=21) inches.

Step3: Use the Pythagorean theorem in right - triangle (DOC)

In right - triangle (DOC), (DC = 25) inches and (DO = 20) inches. By the Pythagorean theorem, (CO=\sqrt{DC^{2}-DO^{2}}). Substitute (DC = 25) and (DO = 20) into the formula: (CO=\sqrt{25^{2}-20^{2}}=\sqrt{(25 + 20)(25 - 20)}=\sqrt{45\times5}=\sqrt{225}=15) inches.

Step4: Calculate the length of (AC)

Since (AC=AO + CO), and (AO = 21) inches and (CO = 15) inches, then (AC=21 + 15=36) inches.

Answer:

36