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.
Math
PopAi provides you with resources such as math solver, math tools, etc.
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.
Start Using PopAi Today
More AI Tools
Suggested Content
More >
What does the sine function represent on the unit circle?
Answer 1 On the unit circle, the sine function $ \sin(\theta) $ represents the y-coordinate of a point on the circle corresponding to the angle $ \theta $ measured from the positive x-axis.\nMathematically, if a point on the unit circle is given by $...
Determine the values of trigonometric functions for specific angles on the unit circle
Answer 1 Given the angle $ \theta = \frac{5\pi}{4} $ radians, find the values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $. The coordinates of $ \frac{5\pi}{4} $ on the unit circle are $ \left(-\frac{\sqrt{2}}{2},...
Find the value of arcsin(1/2) using the unit circle
Answer 1 To find the value of $ \arcsin(\frac{1}{2}) $ using the unit circle, we need to determine the angle whose sine is $ \frac{1}{2} $.On the unit circle, the sine of an angle is the y-coordinate of the corresponding point.The angle that has a...
Determine the coordinates of a point on the unit circle where the sine value is 1/2 and the tangent value is positive
Answer 1 To find the coordinates where $\sin(\theta) = \frac{1}{2}$ and $\tan(\theta)$ is positive, we analyze the unit circle.\n The sine function equals $\frac{1}{2}$ at two angles: $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$.\n Since...
Find the value of tan(θ) using the unit circle when θ is in the third quadrant
Answer 1 To find the value of $ \tan(θ) $ using the unit circle, we need to determine the coordinates where $ θ $ intersects the unit circle in the third quadrant.In the third quadrant, both the x and y coordinates are negative. Suppose $ θ = 225° $...
Find the cosine of the angle pi/4 on the unit circle
Answer 1 The unit circle defines the standard positions and values of trigonometric functions. For the angle $ \frac{\pi}{4} $ (or 45 degrees), we use the unit circle definition:The coordinates of the point on the unit circle corresponding to the...