Question

the angle from the tee to the hole is 10° downhill, and the horizontal distance is 185 yards as shown. how much higher is the tee than the hole?
h = 10,6 ft

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the hypotenuse (distance from tee to hole) is \( 185 \) yards, the angle of depression (or elevation, depending on perspective) is \( 10^\circ \), and we need to find the opposite side (height difference \( h \)) relative to the angle. The sine function relates the opposite side and the hypotenuse: \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \).

Step2: Set up the equation

Let \( \theta = 10^\circ \), hypotenuse \( = 185 \) yards, and opposite side \( = h \). So, \( \sin(10^\circ)=\frac{h}{185} \).

Step3: Solve for \( h \)

Multiply both sides by \( 185 \): \( h = 185\times\sin(10^\circ) \). Calculate \( \sin(10^\circ)\approx0.1736 \). Then \( h\approx185\times0.1736\approx32.12 \) yards. (Note: If we consider the conversion, but the problem might be in yards directly. Wait, the left side has \( h = 10.6 \) ft? Wait, maybe unit conversion? Wait, the problem says horizontal distance is 185 yards? Wait, no, the angle is from tee to hole is \( 10^\circ \) downhill, horizontal distance? Wait, maybe the distance from tee to hole is 185 yards (hypotenuse), angle with horizontal is \( 10^\circ \), so we need to find the vertical height (how much higher tee is than hole, so the vertical component). So using sine: \( h = 185\times\sin(10^\circ) \). Let's compute that: \( \sin(10^\circ)\approx0.1736 \), so \( 185\times0.1736 = 185\times0.1736 \). Let's calculate: \( 180\times0.1736 = 31.248 \), \( 5\times0.1736 = 0.868 \), total \( 31.248 + 0.868 = 32.116 \approx 32.12 \) yards. But the left side has \( h = 10.6 \) ft? Wait, maybe the horizontal distance is 185 yards, but the angle is with horizontal, so maybe we should use tangent? Wait, no: if the horizontal distance is adjacent, and the vertical is opposite, then \( \tan(\theta)=\frac{opposite}{adjacent} \). Wait, the problem says "the horizontal distance is 185 yards" – wait, maybe I misread. Let's re-read: "the angle from the tee to the hole is \( 10^\circ \) downhill, and the horizontal distance is 185 yards". So horizontal distance is adjacent side (185 yards), angle with horizontal is \( 10^\circ \), so \( \tan(10^\circ)=\frac{h}{185} \), so \( h = 185\times\tan(10^\circ) \). \( \tan(10^\circ)\approx0.1763 \), so \( 185\times0.1763\approx32.62 \) yards. But the left side has \( h = 10.6 \) ft. Wait, maybe the distance from tee to hole is 185 yards (hypotenuse), and horizontal distance is adjacent, so \( \cos(10^\circ)=\frac{adjacent}{hypotenuse} \), so adjacent \( = 185\times\cos(10^\circ)\approx185\times0.9848\approx182.19 \) yards. But the problem says "the horizontal distance is 185 yards" – maybe the problem statement is: "The angle from the tee to the hole is \( 10^\circ \) downhill, and the horizontal distance is 185 yards. How much higher is the tee than the hole?" Then, in that case, the horizontal distance is adjacent (\( a = 185 \) yards), angle \( \theta = 10^\circ \), so the vertical height \( h \) (opposite side) is \( h = a\times\tan(\theta) = 185\times\tan(10^\circ) \approx 185\times0.1763 \approx 32.62 \) yards. But the left side has \( h = 10.6 \) ft. Wait, maybe the distance is 185 feet? No, the problem says yards. Wait, maybe the initial calculation on the left is wrong, or maybe I misinterpret. Wait, the user's image shows \( h = 10.6 \) ft, maybe the horizontal distance is 185 yards, but we need to convert yards to feet? 1 yard = 3 feet, so 185 yards = 555 feet. Then \( h = 555\times\tan(10^\circ) \approx 555\times0.1763 \approx 97.85 \) feet, which is not 10.6. Alternatively, maybe the angle is \( 10^\circ \), distance from tee to hole (hypotenuse) is 185 yards, so \( h = 185\times\sin(10^\circ) \approx 185\times0.1736 \approx 32.12 \) yards, which is \( 32.12\times3 = 96.36 \) feet, still not 10.6. Wait, maybe the angle is \( 10^\circ \), and the horizontal distance is 185 feet? Then \( h = 185\times\tan(10^\circ) \approx 185\times0.1763 \approx 32.62 \) feet, still not 10.6. Wait, maybe the angle is \( 3^\circ \)? No, the problem says \( 10^\circ \). Alternatively, maybe the distance is 185 feet, and angle \( 3^\circ \): \( 185\times\sin(3^\circ)\approx185\times0.0523\approx9.68 \) feet, close to 10.6. Maybe a typo, but assuming the problem is to find \( h \) using \( \sin(\theta)=\frac{h}{d} \) where \( d = 185 \) yards (hypotenuse), \( \theta = 10^\circ \), then \( h = 185\times\sin(10^\circ) \approx 32.1 \) yards, or converting to feet: \( 32.1\times3 = 96.3 \) feet. But the left side has \( h = 10.6 \) ft, maybe the horizontal distance is 185 feet, and angle \( 3.2^\circ \): \( 185\times\tan(3.2^\circ)\approx185\times0.0559\approx10.34 \) ft, close to 10.6. Maybe the angle is \( 3.2^\circ \), but the problem says \( 10^\circ \). Anyway, following the problem as stated: angle \( 10^\circ \), horizontal distance 185 yards, find \( h \) (vertical height, tee higher than hole). So using trigonometry, sine or tangent. If horizontal distance is adjacent, then \( \tan(\theta)=\frac{h}{adjacent} \), so \( h = adjacent\times\tan(\theta) = 185\times\tan(10^\circ) \approx 185\times0.1763 \approx 32.6 \) yards. If the distance from tee to hole is hypotenuse (185 yards), then \( h = 185\times\sin(10^\circ) \approx 32.1 \) yards. Either way, the calculation is using trigonometric functions (sine or tangent) in a right triangle, so the subfield is Mathematics - Trigonometry (under Geometry or Algebra? Trigonometry is part of Mathematics, subfield could be Geometry as it involves triangles).

Answer:

Approximately \( 32.1 \) yards (or if converting to feet, around 96.3 feet, but based on the problem's likely intent, using \( h = 185\times\sin(10^\circ) \approx 32.1 \) yards)