Algebra Answer Note #5

Question 1 1. Solve the following linear programming model graphically: Maximize Z=14x + 10y Subject to: 8x + 6y≤480 4x+2y≤200 x,y≥0 Answer Linear programming is a mathematical optimization technique used to determine the best possible outcome based on predetermined constraints. It is commonly used in project management to allocate resources, minimize costs, and maximize profits. In this question, we are required to solve a linear programming model graphically to maximize the objective function Z, while satisfying the given constraints. To solve this problem, we first need to plot the feasible region on a graph. The feasible region is the area where all the constraints are satisfied. In this case, the constraints given are 8x + 6y ≤ 480, 4x + 2y ≤ 200, and x,y ≥ 0. These constraints form a region on the graph as shown in the figure below. [Insert graph here] Next, we need to calculate the coordinates of the vertices of the feasible region. This can be done by solving the equations formed by the two constraints that intersect at a particular point. The vertices of the feasible region are (0,0), (0,80), (25,50), and (60,0). Now, we need to determine the value of the objective function Z at each of these vertices. This can be done by substituting the coordinates into the equation Z = 14x + 10y. The four values obtained are as follows: Z at (0,0) = 0 Z at (0,80) = 800 Z at (25,50) = 900 Z at (60,0) = 840 The highest value of Z is obtained at (25,50), which gives us the maximum profit of $900. Therefore, the optimal solution to this linear programming model is x = 25 and y = 50, with a maximum value of Z = $900. Question 2 Given a =6 i +4 j −4 k and b =3 i +x j −9 k find x when a ⋅ b =−2 Answer First, let's understand the given vectors a and b. Vector a has components of 6i, 4j, and 4k, while vector b has components of 3i, xj, and -9k. We need to find the value of x when a ⋅ b = -2. To solve this, we will use the dot product formula: a ⋅ b = (ax * bx) + (ay * by) + (az * bz). Substituting the values from vectors a and b, we get: (6*x) + (4*x) + (-4*-9) = -2. Simplifying this equation, we get: 10x + 36 = -2. To solve for x, we need to isolate it on one side and simplify. This can be done by subtracting 36 from both sides: 10x = -38. Finally, we divide both sides by 10 to get the value of x: x = -38/10 = -3.8. Therefore, the value of x is -3.8 when a ⋅ b = -2. Question 3 Prove that vector A=(i+3j+k) and vector B=(2i+j+k) are perpendicular to each other Above is the question can you provide the steps also......? Answer Vectors in physics are quantities that have both magnitude and direction. In order to prove that two vectors are perpendicular to each other, we must show that their dot product, also known as scalar product, is equal to zero. The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them. If the dot product is zero, it means that the vectors are perpendicular to each other, as the cosine of a 90-degree angle is equal to zero. Now let’s begin with the proof: Step 1: Finding the dot product of vector A and vector B. We can find the dot product by multiplying their x-components, y-components, and zcomponents and then adding them together. A • B = (i)(2i) + (3j)(j) + (k)(k) =2 + 3 + 1 =6 Step 2: Finding the magnitudes of vector A and vector B. The magnitude of a vector is defined as its length, which can be calculated using the Pythagorean theorem, where the magnitude is the square root of the sum of the squared components. |A| = √(i^2 + 3^2 + 1^2) =√11 ≈3.316 |B| = √(2^2 + 1^2 + 1^2) =√6 ≈2.449 Step 3: Finding the cosine of the angle between vector A and vector B. To find the cosine of the angle between the two vectors, we will use the formula: cosθ = A • B / (|A||B|). cosθ = 6 / (3.316)(2.449) =6 / 8.118 ≈0.738 Step 4: Checking the angle between the two vectors. As we know that the dot product of perpendicular vectors must be zero and the cosine of a 90-degree angle is zero, we can check if the angle between vector A and B is indeed 90 degrees. cos^-1(0.738) ≈46.982 degrees Since the angle between the two vectors is not 90 degrees, we can conclude that the dot product being equal to zero is not a coincidence. Step 5: Conclusion. After evaluating the dot product, magnitudes, and angle between vector A and B, we can conclude that the dot product of the two vectors is not zero, and since the dot product is not equal to zero, it means that the two vectors are not perpendicular to each other.