answer:To find how far up the wall the ladder reaches, we can use the sine function in trigonometry. Let's call the distance from the bottom of the ladder to the wall "x" and the height on the wall where the ladder touches "y". We are given that the ladder makes an angle of 60 degrees with the ground and has a length of 10 meters. Using the sine function, we get: sin(60°) = opposite side (y) / hypotenuse (10 m) Since sin(60°) = √3/2, we can rewrite the equation as: √3/2 = y / 10 Now, we can solve for y: y = 10 * √3/2 y = 5√3 So, the ladder reaches 5√3 meters up the wall.,To find how far up the wall the ladder reaches, we can use the sine function in trigonometry. The sine function relates the angle, the length of the hypotenuse (the ladder in this case), and the length of the side opposite the angle (the height up the wall). sin(angle) = opposite side / hypotenuse In this problem, the angle is 60 degrees, and the hypotenuse (the ladder) is 10 meters long. We want to find the opposite side (the height up the wall). sin(60) = height / 10 To find the sine of 60 degrees, we can either use a calculator or recall that sin(60) = √3/2. (√3/2) = height / 10 Now, we can solve for the height: height = 10 * (√3/2) height = 5√3 Therefore, the ladder reaches approximately 5√3 ≈ 8.66 meters up the wall., To find the distance the ladder reaches up the wall, we can use trigonometry. In this case, we'll use the sine function, which is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In our problem, the ladder acts as the hypotenuse, and the distance up the wall is the opposite side. Let this distance be "a" meters. The adjacent side is the ground distance, which we don't know, but we can use the 60-degree angle and the ladder length to find the opposite side. sin(60°) = a/10 To solve for "a", we first need to find the sine of 60 degrees: sin(60°) = √3 / 2 Now, we can rearrange the equation to solve for "a": a = 10 * sin(60°) a = 10 * (√3 / 2) a ≈ 8.66 meters So, the ladder reaches approximately 8.66 meters up the wall.

answer:To find the angle of depression, we can use the tangent function. The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the altitude of the helicopter (500 meters), and the adjacent side is the horizontal distance to the destination (2000 meters). angle of depression = arctan(opposite / adjacent) angle of depression = arctan(500 / 2000) angle of depression ≈ arctan(0.25) To find the value of the arctan(0.25), we can use a calculator: arctan(0.25) ≈ 0.25 radians To convert radians to degrees, we can use the conversion factor: 1 radian ≈ 57.2958 degrees 0.25 radians × 57.2958 degrees/radian ≈ 14.32 degrees So, the pilot needs to fly on an angle of depression of approximately 14 degrees to reach the destination.,To find the angle of depression, we need to use the tangent function, which is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In this case, the opposite side is the difference in altitude (0 - 500 = -500 meters), and the hypotenuse is the horizontal distance (2000 meters). So, the tangent of the angle of depression is: tan(angle) = opposite/hypotenuse = -500/2000 To find the angle, we take the inverse tangent (arctangent) of this ratio: angle = arctan(-500/2000) Using a calculator, we find that the angle is approximately -26.57 degrees. However, the angle of depression is more conveniently expressed as an angle above the horizontal, not below. Therefore, we need to add 90 degrees to this angle to find the angle of ascent that the pilot needs to take: angle = -26.57 + 90 = 63.43 Rounding to the nearest degree, the pilot should choose an angle of 63 degrees to reach the destination.,To find the angle of depression, we can use the tangent function in trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the altitude (500 meters) and the adjacent side is the horizontal distance (2000 meters). tan(angle) = opposite / adjacent tan(angle) = 500 / 2000 tan(angle) = 0.25 Now, we need to find the angle whose tangent is 0.25. We can do this using the inverse tangent function (arctan or tan^(-1)). angle = arctan(0.25) angle ≈ 14.04 degrees Rounding to the nearest degree, the angle of depression the pilot needs to fly on to reach the destination is approximately 14 degrees.

answer:To find the distance between the base of the ladder and the building, we can use trigonometry. In this case, we'll use the tangent function (tan), which is defined as the ratio of the opposite side to the adjacent side in a right triangle. In our scenario, the ladder acts as the opposite side, and the distance between the ladder and the building is the adjacent side. We have the following information: Angle (A) = 65 degrees Opposite side (O) = 50 feet First, let's find the tangent of the angle: tan(A) = O / Adjacent side Now, rearrange the formula to solve for the adjacent side: Adjacent side = O / tan(A) Next, plug in the known values: Adjacent side = 50 ft / tan(65 degrees) ≈ 29.3 feet So, the base of the ladder is approximately 29.3 feet away from the building.,We can use trigonometry to solve this problem. Let's denote the distance from the base of the ladder to the building as x. Since the angle between the ladder and the ground is 65 degrees, we can set up the following equation using the sine function: sin(65) = opposite side / hypotenuse In this problem, the opposite side is the height of the ladder (h), and the hypotenuse is the distance from the base of the ladder to the building (x). So, we have: sin(65) = h / x We know that the ladder is 50 feet long, so we have another equation: h + x = 50 Now, we can solve this system of two equations with two unknowns. First, we can find the height of the ladder using the sine function: sin(65) = h / x h = x * sin(65) Next, we can substitute this expression for h into the second equation: x * sin(65) + x = 50 Now, we can solve for x: x (1 + sin(65)) = 50 x = 50 / (1 + sin(65)) x ≈ 25.49 feet So, the base of the ladder is approximately 25.49 feet from the building.,To solve this problem, we can use the trigonometric function cosine. The cosine of an angle in a right triangle is the ratio of the adjacent side (the side next to the angle) to the hypotenuse (the longest side, opposite the right angle). In this case, the adjacent side is the distance from the base of the ladder to the building, which we'll call x, and the hypotenuse is the length of the ladder, which is 50 feet. We can set up the equation: cos(65°) = x / 50 Now, we can find the cosine of 65 degrees: cos(65°) ≈ 0.4226 Next, we can solve for x: 0.4226 = x / 50 x ≈ 0.4226 * 50 x ≈ 21.13 So, the base of the ladder is approximately 21.13 feet from the building.

answer:We can use the sine function to solve this problem. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse (the ladder in this case). Let L be the length of the ladder. We have: sin(60°) = (opposite side) / L The opposite side in this case is the distance from the bottom of the ladder to the wall, which is 6 feet. The sine of 60 degrees is equal to √3/2. So we have: (√3/2) = 6 / L Now, we can solve for L: L = 6 / ( √3/2) To get rid of the fraction in the denominator, we can multiply both the numerator and the denominator by 2: L = (6 * 2) / √3 Now, we can simplify: L = 12 / √3 To rationalize the denominator, we can multiply both the numerator and the denominator by √3: L = (12 * √3) / ( √3 * √3) L = (12 * √3) / 3 Finally, we can simplify: L = 4 * √3 So, the length of the ladder is 4√3 feet.,To find the length of the ladder, we can use trigonometry. In this right triangle formed by the ladder, the wall, and the ground, the angle between the ladder and the wall is 60 degrees. We know the length of the adjacent side (the distance from the base of the wall to the ladder) is 6 feet. We want to find the length of the ladder, which is the hypotenuse of the triangle. Using the cosine function, we can set up this equation: cos(60°) = length of ladder / 6 (where cos(60°) = 0.5, as cosine of 60 degrees is 1/2) Now, we solve for the length of the ladder: length of ladder = 6 * cos(60°) = 6 * 0.5 = 3 feet So, the ladder is 3 feet long., To find the length of the ladder, we can use trigonometry and the given angle of 60 degrees. In this case, we'll use the sine function, which is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In our problem, the opposite side represents the height (or distance from the ground to the top of the ladder), the hypotenuse is the ladder's length, and the adjacent side is the distance from the wall to the base of the ladder. Since we're looking for the ladder's length, we can use the sine formula: sin(θ) = (opposite side) / hypotenuse Where θ is the angle between the ladder and the ground (60 degrees in this case). We know the adjacent side (6 feet) and want to find the hypotenuse (the ladder's length). We just need to rearrange the equation to solve for the hypotenuse: hypotenuse = (opposite side) / sin(θ) First, convert the angle from degrees to radians: 60 degrees * (π radians / 180 degrees) ≈ 1.0472 radians Now, plug in the values into the equation: hypotenuse = 6 feet / sin(1.0472 radians) ≈ 6 feet / 0.8660 ≈ 6.928 feet Therefore, the length of the ladder is approximately 6.93 feet (rounded to two decimal places).