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To find the derivatives dz/dx and dz/dy using implicit differentiation from the equation e^(6z) = xyz, we’ll follow these steps: Differentiate both sides: Start with the given equation: e^(6z) = xyz Apply implicit differentiation to both sides with respect to x and then y separately. Differentiate the left-hand side: Using the chain rule, differentiate e^(6z): … Read more

To find dy/dx by implicit differentiation for the equation: x² + xy + y² = 9 we will differentiate both sides of the equation with respect to x. Remember, when differentiating terms that include y, we need to apply the chain rule since y is a function of x. Step 1: Differentiate each term Differentiate … Read more

To find the value of sin 15 degrees, we can utilize the sine subtraction formula, which states: sin(a – b) = sin(a)cos(b) – cos(a)sin(b) For our case, we can set: a = 30 degrees b = 15 degrees This means: sin(15) = sin(30 – 15) Substituting into the formula: sin(15) = sin(30)cos(15) – cos(30)sin(15) We … Read more

To find the measure of angle B in triangle ABC with sides a = 3, b = 5, and c = 7, we can use the Cosine Rule. The Cosine Rule states: c2 = a2 + b2 – 2ab * cos(C) In our case, since we want to find angle B, we can rearrange the … Read more

To solve the differential equation \\( rac{dy}{dx} = x(1 – y^2) \\) using separation of variables, we follow these steps: Separate the variables: We want to rearrange the equation so that all terms involving \\( y \\) are on one side and all terms involving \\( x \\) are on the other side. Starting with … Read more

To solve for b2 in the equation 12h(b1b2), we start by examining the variables and their given values. We know that: a = 16h b1 = 3 Given the equation: 12h(b1b2) = a First, we’ll substitute a with 16h: 12h(b1b2) = 16h Next, we can divide both sides of the equation by h to simplify. … Read more

To solve the problem, we will use the relationship between the variables x and y, defined by the equation y = 2x + 1. (a) Finding dy/dt when dx/dt = 15 and x = 4: We begin with the derivative of y with respect to t: y = 2x + 1 Differentiate both sides with … Read more

To convert the Cartesian equation y = 7 into its polar form, we start by recalling the relationship between Cartesian and polar coordinates. In polar coordinates, the connections are defined as follows: x = r * cos(θ) y = r * sin(θ) In the given equation, we can substitute for y using the polar equivalent: … Read more

In triangle ABC, when angle A is a right angle, it means that angle A measures exactly 90 degrees. In a right triangle, the side that is opposite to the right angle is known as the hypotenuse. Therefore, in this case, the side that is opposite angle A is the hypotenuse of the triangle. To … Read more

To find the probability of the union of two independent events A and B, we can use the following formula: P(A ∪ B) = P(A) + P(B) – P(A ∩ B) Since A and B are independent events, the probability of their intersection can be calculated as: P(A ∩ B) = P(A) × P(B) Now, … Read more