Question

write an equation that can be used to answer the question. then solve.
round to the nearest tenth if necessary. (examples 1 and 2)

1. how far up the tree is the cat?
2. how deep is the water?

find the missing measure in each figure below. round to the nearest tenth
if necessary. (example 3)

Explanation:

Step1: Apply Pythagorean theorem for the first problem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse of a right - triangle. For the cat - tree problem, \(c = 12\) ft, \(a=5\) ft, and \(b = h\). The equation is \(h^{2}+5^{2}=12^{2}\). Then \(h^{2}=12^{2}-5^{2}=144 - 25=119\), so \(h=\sqrt{119}\approx10.9\) ft.

Step2: Apply Pythagorean theorem for the second problem

For the water - depth problem, \(c = 15\) ft, \(a = 6\) ft, and \(b=x\). The equation is \(x^{2}+6^{2}=15^{2}\). Then \(x^{2}=15^{2}-6^{2}=225 - 36 = 189\), so \(x=\sqrt{189}\approx13.7\) ft.

Step3: Apply Pythagorean theorem for the third problem

For the pyramid problem, assume the base half - length. If the base has lengths \(4\) cm and \(8\) cm, the base half - diagonal \(d=\sqrt{4^{2}+(8\div2)^{2}}=\sqrt{16 + 16}=\sqrt{32}\). The equation is \(x^{2}=11^{2}-\sqrt{32}^{2}=121-32 = 89\), so \(x=\sqrt{89}\approx9.4\) cm.

Step4: Apply Pythagorean theorem for the fourth problem

For the cone problem, \(c=x\), \(a = 12\) ft, \(b = 9\) ft. The equation is \(x^{2}=12^{2}+9^{2}=144+81=225\), so \(x = 15\) ft.

Answer:

1. The cat is approximately \(10.9\) ft up the tree.
2. The water is approximately \(13.7\) ft deep.
3. The missing measure \(x\) is approximately \(9.4\) cm.
4. The missing measure \(x\) is \(15\) ft.