What is the height of the airplane when it sights an atoll at an angle of depression of 10 degrees with a horizontal distance of 5172 meters?
To find the height of the airplane, we can use trigonometric functions, specifically the tangent function. The angle of depression corresponds to the angle made with the horizontal line from the airplane to the atoll. 1. The scenario can be visualized as a right triangle where: The angle of depression is 10 degrees. The horizontal […]
Find the coordinates of the midpoint of the segment whose endpoints are (6, 4) and (2, 8)
To find the midpoint of a line segment defined by two endpoints, you can use the midpoint formula. The formula is given by: Midpoint (M) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2) Here, (x₁, y₁) and (x₂, y₂) are the coordinates of the two endpoints. In this case, the endpoints are: […]
How do you solve the system of equations 2x + y = 3 and x + 2y = 1?
To solve the system of equations: 2x + y = 3 (1) x + 2y = 1 (2) We can use the method of substitution or elimination. Here, we’ll use the elimination method. First, we can rearrange equation (1) for y: y = 3 – 2x Next, we substitute this expression for y into equation […]
For which value of m does the graph of y = 18x² + mx + 2 have exactly one x-intercept?
To determine the value of m for which the quadratic equation y = 18x² + mx + 2 has exactly one x-intercept, we need to consider the nature of the roots of the equation. A quadratic equation will have exactly one x-intercept if its discriminant is equal to zero. The discriminant (D) for a quadratic […]
Which linear function represents the line given by the point-slope equation y – 7 = 2/3 (x – 6)?
To find the linear function represented by the point-slope equation y – 7 = 2/3 (x – 6), we need to transform this equation into the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Start by distributing the 2/3 on the right side: y […]
Vectors A, B, and C: Finding the Angles Between Them
To find the angles between vectors A and B, A and C, and B and C, we can use the dot product formula: Cosine of the angle: cos(θ) = (A • B) / (|A| |B|) Here’s how to approach it: Calculate the dot product: Find the dot product A • B, A • C, and […]
How do you solve for x in the equation 3x^4y^8 = 6x^7y^2?
To solve the equation 3x4y8 = 6x7y2, we will start by simplifying each side of the equation. First, we can divide both sides by y2 (assuming y ≠ 0), which gives us: 3x4y6 = 6×7 Next, we can also divide both sides by 3: x4y6 = 2×7 Now, we can rearrange the equation to isolate […]
How to Find the Length of the Curve r(t) = 5t, 3cos(t), 3sin(t) at t = 3?
To find the length of the curve given by the parametric equations r(t) = (5t, 3cos(t), 3sin(t)), we will use the formula for the length of a curve. The length L of a curve defined by parametric equations from t = a to t = b is given by: L = ∫ab √((dx/dt)² + (dy/dt)² […]
Give a recursive definition of the sequence a_n = 2n for n = 2, 3, 4 where a_1 = 2
The sequence can be defined recursively by specifying the first term and the relationship between successive terms. Let’s define the sequence {a_n} recursively as follows: Base case: a_1 = 2 Recursive case: a_n = 2 * n for n ≥ 2 Thus, we have: a_1 = 2 a_2 = 2 * 2 = 4 a_3 […]
Choose the solutions of the following system of equations: x² + y² = 6 and x² + y = 6
To solve the system of equations: x² + y² = 6 x² + y = 6 We can start by isolating one variable in one of the equations. Let’s take the second equation: y = 6 – x² Now we can substitute this expression for y into the first equation: x² + (6 – x²)² […]