How do you simplify the expression with both multi-step operations and fraction coefficients: 2/3x – 5/6 = 1/2( x + 10 )?To simplify the expression 2/3x – 5/6 = 1/2(x + 10), first distribute the 1/2 on the right side: 2/3x – 5/6 = 1/2x + 5. Then, to eliminate the fractions, multiply every term by 6: 4x – 5 = 3x + 30. Finally, solve for x by isolating it: x = 35.
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How do you derive the formula for the foci of an ellipse given the lengths of the major and minor axes?
How do you derive the formula for the foci of an ellipse given the lengths of the major and minor axes?To derive the formula for the foci of an ellipse, start with the standard form of the ellipse equation: (x^2/a^2) + (y^2/b^2) = 1, where a is the semi-major axis and b is the semi-minor axis. The distance from the center to each focus (c) is given by the formula c = √(a^2 – b^2).
What is the difference between an integer and a whole number?
What is the difference between an integer and a whole number?An integer is any number from the set of positive and negative whole numbers, including zero (e.g., -3, 0, 4). Whole numbers are a subset of integers that include only non-negative numbers (e.g., 0, 1, 2, 3). Thus, all whole numbers are integers, but not all integers are whole numbers.
How do you differentiate between a population and a sample in statistics, and why is it important to understand the difference?
How do you differentiate between a population and a sample in statistics, and why is it important to understand the difference?In statistics, a population refers to the entire group about which data is being collected, while a sample is a subset of the population. Understanding the difference is crucial because it affects the accuracy and generalizability of statistical inferences. Sampling allows for manageable data collection and analysis, but it also introduces sampling error.
How do you derive the equation of an ellipse given its major and minor axes lengths as well as its foci coordinates, and prove that the distance sum of a point to its foci is constant?
How do you derive the equation of an ellipse given its major and minor axes lengths as well as its foci coordinates, and prove that the distance sum of a point to its foci is constant?Given the lengths of the major axis (2a) and minor axis (2b), and the coordinates of the foci at (±c, 0), the standard form of the ellipse equation is (x²/a²) + (y²/b²) = 1. To prove the distance sum is constant, note that for any point (x, y) on the ellipse, the sum of distances to the foci (d1 + d2) equals 2a, which is constant.
How do you calculate the area of a polygon using the coordinates of its vertices?
How do you calculate the area of a polygon using the coordinates of its vertices?To calculate the area of a polygon using the coordinates of its vertices, apply the Shoelace formula. For a polygon with vertices (x1, y1), (x2, y2), …, (xn, yn), the area A is given by: A = 0.5 * | Σ (xi * yi+1 – xi+1 * yi) |, where the summation is from i = 1 to n, and (xn+1, yn+1) is (x1, y1).
How do you derive the double-angle formulas for sine, cosine, and tangent, and how can these be applied to solve identities and equations in non-right triangles?
How do you derive the double-angle formulas for sine, cosine, and tangent, and how can these be applied to solve identities and equations in non-right triangles?The double-angle formulas are derived from the sum formulas for sine and cosine. For sine: sin(2θ) = 2sin(θ)cos(θ). For cosine: cos(2θ) = cos²(θ) – sin²(θ) or cos(2θ) = 2cos²(θ) – 1 or cos(2θ) = 1 – 2sin²(θ). For tangent: tan(2θ) = 2tan(θ) / (1 – tan²(θ)). These formulas are crucial in solving trigonometric identities and equations in non-right triangles, particularly in the Law of Sines and Law of Cosines, which are used to find unknown sides and angles.
How do you solve equations with both addition and multiplication, for example, how do you solve 2(x + 3) = 14?
How do you solve equations with both addition and multiplication, for example, how do you solve 2(x + 3) = 14?To solve 2(x + 3) = 14, first distribute the 2: 2x + 6 = 14. Subtract 6 from both sides: 2x = 8. Finally, divide by 2: x = 4.
An orchard has a certain number of trees, each tree expected to bear a specific average number of fruits. If the orchard has been measured to occupy ‘x’ percent more area in the past three years, and currently boasts ‘y’ percent more in fruit yield under
An orchard has a certain number of trees, each tree expected to bear a specific average number of fruits. If the orchard has been measured to occupy ‘x’ percent more area in the past three years, and currently boasts ‘y’ percent more in fruit yield under Starting with 250 trees producing 195 fruits each, the initial yield is 48,750 fruits. With ‘y’ percent increase in yield, the new yield is 48,750 * (1 + y/100). Adjusting for yield proportion due to ‘t’ square footage growth, the final yield is this value modified by the growth factor.
How do I solve and graph the inverse of a trigonometric function?
How do I solve and graph the inverse of a trigonometric function?To solve and graph the inverse of a trigonometric function, first restrict the domain of the original function to make it one-to-one. Then, swap the x and y variables to find the inverse. Finally, plot the resulting function, ensuring the graph reflects the inverse relationship correctly.
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Answer 1 To find the value of $ \tan(\frac{\pi}{4}) $ using the unit circle:On the unit circle, the coordinates for $ \frac{\pi}{4} $ are $ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $.Therefore:$ \tan(\frac{\pi}{4}) =...
Find the tangent line equations for every point on the unit circle
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Find the point on the unit circle where the sine value is negative and the cosine value is positive
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Determine the coordinates of points on the unit circle that satisfy the given equation
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