Question

1. a 10 - meter cable is stretched from the top of a pole to an anchor on the ground. it is anchored on the ground 4 meters away from the base of the pole. how tall is the pole?
2. the diagonal of a rectangle measures 12 inches. the width of the rectangle is 6 inches. what is the perimeter of the rectangle?
3. a square has an area of 225 m². what is the length of the squares diagonal?
4. jessicas cat is stuck in a tree. the fire department no longer assists in getting cats out of trees. jessicas dad knows the cat is approximately 22 feet high. he has a 25 - foot ladder and the directions say to be safe he must keep the base of the ladder 10 feet from the base of the tree. will the ladder reach the cat so he can safely get it out of the tree?
5. jake drives a truck with a slide - out ramp for loading motorcycles. the tailgate of his truck is 2 meters above the ground. the ramp is 4 meters long. what is the horizontal distance the ramp can reach?
6. talia needs to paint a 9.5 foot metal rod. she wants to place it on a tarp so the paint does not drip on the floor. she has a rectangular tarp that is 6 feet by 8 feet. will the metal rod fit on the tarp or does she need to buy a new tarp for the project?

Explanation:

Step1: Identify the right - triangle relationship

For problem 5, we have a right - triangle where the cable is the hypotenuse \(c = 10\) meters and the base \(a = 4\) meters. We use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), and we want to find the height of the pole \(b\).
\[b=\sqrt{c^{2}-a^{2}}\]
\[b=\sqrt{10^{2}-4^{2}}=\sqrt{100 - 16}=\sqrt{84}=2\sqrt{21}\approx9.17\] meters

Step2: For problem 6

Let the diagonal of the rectangle be \(d = 12\) inches and the width \(w = 6\) inches. Using the Pythagorean theorem in the rectangle (\(d^{2}=w^{2}+l^{2}\)), we find the length \(l\).
\[l=\sqrt{d^{2}-w^{2}}=\sqrt{12^{2}-6^{2}}=\sqrt{144 - 36}=\sqrt{108}=6\sqrt{3}\] inches
The perimeter \(P=2(l + w)=2(6\sqrt{3}+6)=12(1 + \sqrt{3})\approx12\times(1 + 1.732)=32.784\) inches

Step3: For problem 7

If the area of the square \(A = s^{2}=225\ m^{2}\), then the side - length \(s=\sqrt{225}=15\ m\). Using the Pythagorean theorem for the square (diagonal \(D\) and side - length \(s\)), \(D=\sqrt{s^{2}+s^{2}}=\sqrt{2s^{2}}=s\sqrt{2}\). So \(D = 15\sqrt{2}\approx21.21\) m

Step4: For problem 8

We have a right - triangle with ladder length \(L = 25\) feet as the hypotenuse and base \(b = 10\) feet. We find the height \(h\) the ladder can reach using the Pythagorean theorem \(h=\sqrt{L^{2}-b^{2}}=\sqrt{25^{2}-10^{2}}=\sqrt{625 - 100}=\sqrt{525}=5\sqrt{21}\approx22.91\) feet. Since \(22.91>22\), the ladder will reach the cat.

Step5: For problem 9

We have a right - triangle with ramp length \(r = 4\) meters as the hypotenuse and height \(h = 2\) meters. Using the Pythagorean theorem to find the horizontal distance \(x\), \(x=\sqrt{r^{2}-h^{2}}=\sqrt{4^{2}-2^{2}}=\sqrt{16 - 4}=\sqrt{12}=2\sqrt{3}\approx3.46\) meters

Step6: For problem 10

The diagonal of the rectangular tarp with length \(l = 8\) feet and width \(w = 6\) feet can be found using the Pythagorean theorem \(D=\sqrt{l^{2}+w^{2}}=\sqrt{8^{2}+6^{2}}=\sqrt{64 + 36}=\sqrt{100}=10\) feet. Since \(10>9.5\), the metal rod will fit on the tarp.

Answer:

1. The height of the pole is approximately \(9.17\) meters.
2. The perimeter of the rectangle is approximately \(32.784\) inches.
3. The length of the square's diagonal is approximately \(21.21\) m.
4. Yes, the ladder will reach the cat.
5. The horizontal distance the ramp can reach is approximately \(3.46\) meters.
6. The metal rod will fit on the tarp.