To calculate the run (horizontal distance) of a road with a given grade and rise, you can use the formula for the slope of a right triangle. The grade percentage represents the ratio of the rise (vertical change) to the run (horizontal change). First, let’s understand the relationship between the grade, rise, and run. The … Read more
Finding All Solutions for the Equation 0 = 2p cos x sin x To find all solutions to the equation 0 = 2p cos x sin x within the interval [0, 2π], let’s analyze the equation step by step. Step 1: Simplify the Equation The original equation can be factored as: 0 = 2p cos … Read more
To understand the conversion between cusec (cubic feet per second) and litres, we first need to know what a cusec measures. Cusec is a flow rate measurement, indicating how many cubic feet of water pass through a point in one second. Now, let’s break it down: 1 cubic foot is approximately equal to 28.3168 litres. … Read more
To begin, we recognize that c and d vary inversely. This means that as one variable increases, the other decreases in such a way that their product remains constant. In terms of an equation, we express this relationship as: c * d = k where k is a constant. Given that d = 2 when … Read more
To find the value of k in the given quadratic equation kx² – abx + ab when the roots are 1 and b, we can utilize Vieta’s formulas, which relate the coefficients of a polynomial to sums and products of its roots. The quadratic equation can be expressed in the standard form as: k(x – … Read more
To find the sine, cosine, and tangent of the angle 5π/4 radians, we first need to identify which quadrant this angle lies in. The angle 5π/4 can be converted to degrees for better understanding: 5π/4 radians = (5 × 180°)/4π = 225°. This angle is located in the third quadrant, where both sine and cosine … Read more
To solve the linear equation 10 + 2d + 7 = 8 + 10 + 3d, we will follow these steps: Simplify both sides of the equation: On the left side, combine like terms: 10 + 7 = 17, so we have 17 + 2d. On the right side, combine like terms: 8 + 10 … Read more
Calculating the Area and Perimeter of a Square A square is a special type of polygon that has four equal sides and four right angles. To find the area and perimeter of a square, you can rely on simple formulas that involve its sides. 1. Finding the Area The area of a square is calculated … Read more
To evaluate the surface integral of the paraboloid y = x² + z² that lies inside the cylinder x² + z² = 1, we can follow these steps: Step 1: Set Up the Surface Integral The surface integral we aim to evaluate is given by: ∬_S f(x, y, z) \, dS, where S is the … Read more
Finding the Solution Set of the Equation To solve the equation 8x + 15 = 7x + 12, we can follow these steps: Isolate the variable: Start by getting all terms containing x on one side and constant terms on the other. Subtract 7x from both sides: 8x + 15 – 7x = 7x + … Read more