+115 
Question Number 2, I solved one.
The answer according to the key is: 12.9M
My main issue with this question is interpertation.
+9481 This is an especially tricky question because they do not give us an exact triangle ABC.
Here is a graph to see different possible scenarios: https://www.desmos.com/calculator/t4tcx4ev8m
We can change the length of reachable fence by moving the points so that part of AC is inside the circle without violating any of the rules given in the problem statement. So I would venture to say this is a bad question. I will just make the significant assumption that no part of AC is inside the circle.
Here is a diagram with new points D, E, and F labeled as shown:
length of reachable fence = diameter of circle P + length of chord EF
Let all lengths be in meters.
diameter of circle P = 2 * radius of circle P = 2 * 3.8 = 7.6
| Let's look at △BDP . | ____ | |||
| sin 37° = DP / BP | because △BDP is a right triangle and m∠DBP = 37° |
| ||
| sin 37° = DP / 4.5 | because BP = 4.5 | |||
| 4.5 sin 37° = DP |
|
| ||
| Now let's look at △DEP . |
|
| ||
| ( DE )2 + ( DP )2 = ( EP )2 |
| by the Pythagorean Theorem. |
| |
| ( DE )2 + ( 4.5 sin 37°)2 = ( 3.8 )2 |
| because DP = 4.5 sin 37° and EP = radius of circle P = 3.8 |
| |
| ( DE )2 = ( 3.8 )2 - ( 4.5 sin 37°)2 |
| |||
| DE = √[ ( 3.8 )2 - ( 4.5 sin 37°)2 ] |
|
| ||
| DE ≈ 2.67 | ||||
And by the hypotenuse-leg congruence theorem, △DEP ≅ △DFP . So DE = DF
length of chord EF = DE + DF = DE + DE = 2 * DE ≈ 2 * 2.67 ≈ 5.34
length of reachable fence = diameter of circle P + length of chord EF ≈ 7.6 + 5.34 ≈ 12.9
+9481 This is an especially tricky question because they do not give us an exact triangle ABC.
Here is a graph to see different possible scenarios: https://www.desmos.com/calculator/t4tcx4ev8m
We can change the length of reachable fence by moving the points so that part of AC is inside the circle without violating any of the rules given in the problem statement. So I would venture to say this is a bad question. I will just make the significant assumption that no part of AC is inside the circle.
Here is a diagram with new points D, E, and F labeled as shown:
length of reachable fence = diameter of circle P + length of chord EF
Let all lengths be in meters.
diameter of circle P = 2 * radius of circle P = 2 * 3.8 = 7.6
| Let's look at △BDP . | ____ | |||
| sin 37° = DP / BP | because △BDP is a right triangle and m∠DBP = 37° |
| ||
| sin 37° = DP / 4.5 | because BP = 4.5 | |||
| 4.5 sin 37° = DP |
|
| ||
| Now let's look at △DEP . |
|
| ||
| ( DE )2 + ( DP )2 = ( EP )2 |
| by the Pythagorean Theorem. |
| |
| ( DE )2 + ( 4.5 sin 37°)2 = ( 3.8 )2 |
| because DP = 4.5 sin 37° and EP = radius of circle P = 3.8 |
| |
| ( DE )2 = ( 3.8 )2 - ( 4.5 sin 37°)2 |
| |||
| DE = √[ ( 3.8 )2 - ( 4.5 sin 37°)2 ] |
|
| ||
| DE ≈ 2.67 | ||||
And by the hypotenuse-leg congruence theorem, △DEP ≅ △DFP . So DE = DF
length of chord EF = DE + DF = DE + DE = 2 * DE ≈ 2 * 2.67 ≈ 5.34
length of reachable fence = diameter of circle P + length of chord EF ≈ 7.6 + 5.34 ≈ 12.9
