In a system where you have a sphere inscribed within a right circular cylinder and another sphere that circumscribes this cylinder, if the sphere inscribed in the cylinder has a radius r, find the ratio of the volume of the larger sphere to the volume of The sphere inscribed in the cylinder has a radius r and thus a volume of (4/3)πr^3. The larger sphere circumscribing the cylinder has a diameter equal to the cylinder’s diagonal, which is 2√2r, giving it a radius of √2r and a volume of (4/3)π(√2r)^3. The ratio of the volumes is (√2)^3 = 2√2.
Homework
PopAi provides you with resources such as science, math, humanities, etc.
How does the implementation of a Convolutional Neural Network (CNN) differ from a Recurrent Neural Network (RNN) when processing images and sequential data, respectively?
How does the implementation of a Convolutional Neural Network (CNN) differ from a Recurrent Neural Network (RNN) when processing images and sequential data, respectively?CNNs are designed for spatial data like images, using convolutional layers to detect patterns. RNNs, on the other hand, are tailored for sequential data, utilizing recurrent connections to maintain temporal dependencies. CNNs excel in image recognition tasks, while RNNs are ideal for tasks like language modeling and time series prediction.
If four-fifths of a number is increased by ten, the result is five more than three-fourths of the original number. What is the original number?
If four-fifths of a number is increased by ten, the result is five more than three-fourths of the original number. What is the original number?Let the original number be x. According to the problem, (4/5)x + 10 = (3/4)x + 5. Solving this equation, we find that x = 100. Therefore, the original number is 100.
What are the main functions and responsibilities of the three branches of the United States government?
What are the main functions and responsibilities of the three branches of the United States government?The United States government is divided into three branches: the legislative, executive, and judicial. The legislative branch, composed of Congress, makes laws. The executive branch, led by the President, enforces laws. The judicial branch, headed by the Supreme Court, interprets laws and ensures they align with the Constitution.
How does the law of conservation of mass apply to chemical reactions, and can you provide an example demonstrating this law?
How does the law of conservation of mass apply to chemical reactions, and can you provide an example demonstrating this law?The law of conservation of mass states that mass is neither created nor destroyed in a chemical reaction. This means the mass of the reactants equals the mass of the products. For example, in the reaction 2H₂ + O₂ → 2H₂O, the mass of hydrogen and oxygen before the reaction is equal to the mass of water produced.
How can we differentiate between Type I and Type II errors, and which degree of error is more acceptable in medical testing outcomes for rare diseases?
How can we differentiate between Type I and Type II errors, and which degree of error is more acceptable in medical testing outcomes for rare diseases?Type I error, or false positive, occurs when a test incorrectly indicates the presence of a condition. Type II error, or false negative, occurs when a test fails to detect a condition. In medical testing for rare diseases, minimizing Type II errors is generally more acceptable, as missing a diagnosis can have more severe consequences than a false positive.
How do you solve a system of nonlinear equations involving a quadratic and a linear equation using the substitution or elimination method?
How do you solve a system of nonlinear equations involving a quadratic and a linear equation using the substitution or elimination method?To solve a system of nonlinear equations involving a quadratic and a linear equation using substitution, first solve the linear equation for one variable. Then substitute this expression into the quadratic equation. Solve the resulting quadratic equation for the variable and back-substitute to find the other variable. For elimination, align both equations and eliminate one variable by adding or subtracting the equations, then solve the resulting equation.
How can the Law of Sines be used to determine the radius of the circumscribed circle around a triangle given all three sides of the triangle?
How can the Law of Sines be used to determine the radius of the circumscribed circle around a triangle given all three sides of the triangle?To determine the radius (R) of the circumscribed circle around a triangle using the Law of Sines, use the formula R = a / (2 * sin(A)), where ‘a’ is a side of the triangle and ‘A’ is the angle opposite to it. First, find the angles using the sides and the Law of Cosines, then apply the Law of Sines.
How do you find the zeros of a polynomial function using the Rational Root Theorem and synthetic division?
How do you find the zeros of a polynomial function using the Rational Root Theorem and synthetic division?To find the zeros of a polynomial function using the Rational Root Theorem and synthetic division, first list all possible rational roots using the theorem. Then, use synthetic division to test each candidate root. If the remainder is zero, the candidate is a root. Repeat the process with the quotient polynomial until all zeros are found.
What are the key differences between linear and quadratic functions?
What are the key differences between linear and quadratic functions?Linear functions have the form f(x) = mx + b, where the graph is a straight line, and the rate of change is constant. Quadratic functions have the form f(x) = ax^2 + bx + c, where the graph is a parabola, and the rate of change varies, creating a curved shape.
Start Using PopAi Today
More AI Tools
Suggested Content
More >
Find the equation of a tangent to the unit circle at a given point
Answer 1 To find the equation of a tangent to the unit circle at the point $(a, b)$, we start by noting that the unit circle is defined by: $x^2 + y^2 = 1$ The slope of the radius at $(a, b)$ is $ \x0crac{b}{a} $, so the slope of the tangent line,...
Find the values of tan(θ) for θ in the unit circle at 0, π/4, π/3, and π/2
Answer 1 To determine the values of $ \tan(\theta) $ for $ \theta $ in the unit circle at $ 0 $, $ \frac{\pi}{4} $, $ \frac{\pi}{3} $, and $ \frac{\pi}{2} $, we evaluate the tangent function at these angles:For $ \theta = 0 $:$ \tan(0) = 0 $For $...
Find the angle $ \theta $ on the unit circle where the following conditions are met: $ \sin(\theta) = -\frac{1}{2} $ and $ \cos(\theta) = -\frac{\sqrt{3}}{2} $
Answer 1 To find the angle $ \theta $ on the unit circle where $ \sin(\theta) = -\frac{1}{2} $ and $ \cos(\theta) = -\frac{\sqrt{3}}{2} $, we need to identify the corresponding angles in degrees.First, note that $ \sin(\theta) = -\frac{1}{2} $ occurs...
Find the value of csc(θ) when θ = π/6 on the unit circle
Answer 1 To find the value of $csc(θ)$ when $θ = \frac{π}{6}$ on the unit circle, we first find the sine of $θ$:$ \sin(\frac{π}{6}) = \frac{1}{2} $Since $csc(θ) = \frac{1}{\sin(θ)}$, we have:$ csc(θ) = \frac{1}{\sin(\frac{π}{6})} =...
Find the exact value of cos(pi/9) using the unit circle and trigonometric identities
Answer 1 To find the exact value of $ \cos(\frac{\pi}{9}) $, we can use the triple-angle identity for cosine:\n$ \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) $\nLetting $ \theta = \frac{\pi}{9} $, we get:\n$ \cos(\frac{\pi}{3}) =...
Evaluate the integral of sec(x) along the unit circle
Answer 1 To evaluate the integral of $ \sec(x) $ along the unit circle, we consider the parametrization of the unit circle. The unit circle can be parametrized as $ x = \cos(\theta) $ and $ y = \sin(\theta) $, where $ \theta $ ranges from $ 0 $ to $...