Question

the lacrosse player shown passes the ball to a team member who is directly in front of the center of the net and is directly below her in the diagram. if the team member catches the ball and then throws it to the center of the net, what is the total distance of the two throws? the corners of the net are at (15, 27) and (15, 33).

Explanation:

Step1: Find the center of the net

The corners of the net are at \((15, 27)\) and \((15, 33)\). Since they have the same \(x\)-coordinate, the center of the net will have the same \(x\)-coordinate (\(15\)) and the \(y\)-coordinate will be the average of \(27\) and \(33\).
\[ y_{\text{center}}=\frac{27 + 33}{2}=\frac{60}{2}=30 \]
So the center of the net is at \((15, 30)\).

Step2: Find the coordinates of the team member

The team member is directly in front of the center of the net (so same \(x\)-coordinate as the center of the net, \(x = 15\)) and directly below the lacrosse player. The lacrosse player is at \((45, 50)\), so the team member has the same \(x\)-coordinate as the center of the net (\(15\))? Wait, no, wait. Wait, "directly in front of the center of the net" and "directly below her (the lacrosse player)". Wait, the lacrosse player is at \((45, 50)\). "Directly below" would mean same \(x\)-coordinate? Wait, no, maybe "directly below" in the diagram means same \(x\)-coordinate? Wait, no, the center of the net is at \((15, 30)\)? Wait, no, wait the corners of the net are at \((15, 27)\) and \((15, 33)\), so the net is along the line \(x = 15\), from \(y = 27\) to \(y = 33\). The team member is directly in front of the center of the net (so in front of the net, so same \(x\)-coordinate as the center of the net? Wait, no, "in front of the center of the net" – maybe the net is at \(x = 15\), and the field is from \(x = 0\) to \(x = 100\)? Wait, the diagram shows 100 yards width. Wait, the lacrosse player is at \((45, 50)\). The team member is directly below her, so same \(x\)-coordinate (\(45\)) and directly in front of the center of the net. Wait, maybe I misread. Let's re - examine:

Wait, the center of the net: since the two corners are \((15, 27)\) and \((15, 33)\), the center is \((15,\frac{27 + 33}{2})=(15, 30)\). The team member is directly in front of the center of the net (so same \(y\)-coordinate as the center of the net? No, "directly in front of the center of the net" – maybe the net is at the back ( \(x = 15\) is the back - line?), and the field is from \(x = 15\) to \(x = 100\)? Wait, the diagram has a 15 - yard arrow towards the net. Wait, maybe the lacrosse player is at \((45, 50)\), the team member is directly below her (so same \(x = 45\)) and directly in front of the center of the net (so same \(y\) as the center of the net? No, the center of the net is at \((15, 30)\). Wait, no, maybe "directly in front of the center of the net" means that the team member is on the line perpendicular to the net (which is vertical, since net corners are \((15,27)\) and \((15,33)\)) so horizontal line. And "directly below the lacrosse player" means same \(x\)-coordinate as the lacrosse player. So the lacrosse player is at \((45, 50)\), so the team member is at \((45, 30)\) (since center of net is at \((15, 30)\)? No, that can't be. Wait, maybe the net is at \(x = 15\), and the team member is at \((45, 30)\) (same \(y\) as center of net, same \(x\) as lacrosse player? No, the problem says "directly in front of the center of the net and is directly below her in the diagram". So "directly below her" – same \(x\)-coordinate as the lacrosse player (\(x = 45\)), and "directly in front of the center of the net" – same \(y\)-coordinate as the center of the net. The center of the net is at \((15, 30)\)? No, that would mean the team member is at \((45, 30)\). Let's check:

First throw: from lacrosse player \((45, 50)\) to team member \((45, 30)\). The distance is \(|50 - 30|=20\) yards (since same \(x\)-coordinate, distance is difference in \(y\)-coordinates).

Second throw: from team member \((45, 30)\) to center of net \((15, 30)\). The distance is \(|45 - 15| = 30\) yards (since same \(y\)-coordinate, distance is difference in \(x\)-coordinates).

Total distance: \(20+30 = 50\) yards. Wait, let's verify the center of the net again. The two corners are \((15,27)\) and \((15,33)\). The mid - point (center) is \((15,\frac{27 + 33}{2})=(15, 30)\), correct. The lacrosse player is at \((45, 50)\). The team member is directly below her (so \(x = 45\)) and directly in front of the center of the net (so \(y = 30\)), so team member is at \((45, 30)\).

First distance: between \((45, 50)\) and \((45, 30)\): \(d_1=\sqrt{(45 - 45)^2+(50 - 30)^2}=\sqrt{0 + 400}=20\)

Second distance: between \((45, 30)\) and \((15, 30)\): \(d_2=\sqrt{(45 - 15)^2+(30 - 30)^2}=\sqrt{900+0}=30\)

Total distance: \(d_1 + d_2=20 + 30=50\)

Answer:

The total distance of the two throws is \(\boldsymbol{50}\) yards.