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This document provides step-by-step working through solving the equation cos^2θ = sin^2θ to determine the value of theta. It starts by rearranging and factoring the equation, then applying the Pythagorean identity to obtain sin^2θ = 1. Taking the reciprocal of both sides and manipulating further isolates sinθ, revealing that sinθ = 1 and thus θ = 90 degrees or π/2 radians. The summary concludes by noting the general solutions for θ in degrees and radians where sinθ = 1.
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Problem 3
The derivative of the function f(x) = 3(x-2)^2 + 2 is 6(x-2). The equation of the tangent line to f(x) when x is 4 is y = 18 + 6(x - 4). Given that g(x)=√x and the points of intersection between the tangent lines to f(x) at x=5 and g(x) at x=4 are the centers of three circles with radius 5, the equations of the circles are (x-c1)2 + (y-d1)2 = 25, (x-c2)2 + (y-d2)2 = 25, (x-c3)2
Solution 3
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.
Problem 1
The problem asks to find the area bounded by the axes and the curves y=12 and the perimeter of the conic section with the equation (x-15)^2 – r^2 = -(y-12)^2, where r is the positive square root of D rounded to the nearest whole number. The document provides the equation to determine the value of D needed for the conic section equation.
Solution 1
The document presents a multi-step problem to determine the area bounded by a circle, the x-axis, y-axis, and the line y=12. It provides the equation of the circle and guides the reader through simplifying systems of equations to find values for variables in the circle equation. These are used to graph the circle and identify the bounded area as a rectangle minus one quarter of the circle area, calculated as 66.9027 square units.
Problem 4
You are very tired working on a late project due today at 3:25 AM. You still need one more question but your mind is weakened from lack of sleep and stress. Rather than continuing fruitless efforts, you decide to stop working and begin reviewing for an approaching math exam, knowing it is a higher priority. However, when you look at a review question you freeze, not knowing how to solve it.
Solution 4
The document summarizes the step-by-step working of a math problem to determine the value of theta given a trigonometric equation. It rearranges the equation, uses trigonometric identities like the Pythagorean identity, isolates the sine term, and determines that theta equals 90 degrees or pi/2 radians based on the unit circle. It then provides the general form of all solutions for theta in both degrees and radians.
Problem 4
The student is very tired and stressed trying to finish a project that is deterring from studying for exams. It is 3:25 AM and they still need to come up with one final question for the project. Their mind is weakened from lack of sleep and stress. Rather than continuing to struggle on the project question, they decide to stop and begin reviewing for an approaching math exam, knowing it is more important. However, when they look at a review question, they freeze up from not knowing how to solve it.
Solution 3
The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.
Problem 3
The derivative of the function f(x) = 3(x-2)^2 + 2 is 6(x-2). The equation of the tangent line to f(x) when x is 4 is y = 18 + 6(x - 4). Given the functions f(x) = 3(x-2)^2 + 2 and g(x) = sqrt(x), the points of intersection between their tangent lines at x=2 and x=5 determine the centers of three circles with radius 5.
Problem 2
Emar is on the second floor of a building and begins walking continuously up and down the stairway that connects the two floors. The stairway has a slope defined by an equation and it takes Emar 75 seconds to travel between floors. As he walks, his height follows a sinusoidal pattern over time. Sophia is also on the stairway and is moving down at a steady velocity. The question asks to determine the equation for Emar's height over time, calculate the height the two will be at the same spot, and count the possible combinations of selected people at a given time.
Solution 1
The document provides the equation of a conic section and asks to determine the area bounded by the conic section, the x-axis, y-axis, and the line y=12. It involves solving a system of equations to find values for variables in the conic equation, then using these values to graph the circle and calculate the bounded area. The area is found to be 66.9027 square units.
Problem 1
To find the area bounded by the axes and the curves y=12 and the conic section with equation (x-15)^2 – r^2 = -(y-12)^2, first determine D using the provided equations, take the positive square root of D and round to the nearest whole number to find r, then use the conic section equation to find the perimeter and bound the area.
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Solution 4
- 1. Intro. To Problem 4 You are very tired. You are doing a project that is deterring your efforts to study for exams. You feel that you will not be prepared for exams because of that certain project. Today is the due date of your project and it is 3:25 AM. You still need to think of one question. Then after that, you’re going to need to provide an explanation for it and post it on your website to showcase your project. You keep thinking to yourself, “What should the final question be?” You come to understand that you’re mind has been weakened from lack of sleep, stress, and too much partying. Even great minds can’t always think. What you do is cease your fruitless efforts on your project and review for math because you understand that the provincial math exam is approaching in about 4 days. You grab some sheets and begin reading. You look at one certain question. Just by looking at it, you freeze up from not knowing how to answer it. Well well, let’s take a look at that question again… Muahaha!!!
- 2. Elegant Problem Determine Theta Given The Equation Below: =
- 3. Elegant Problem Determine Theta Given The Equation Below: = This is the last question, which is the most simple looking of the four, yet the question itself requires lots of thinking. Let’s start by rearranging the equation. After the rearrangement, you should notice something. =
- 4. Elegant Problem Determine Theta Given The Equation Below: = Maybe the brackets help you notice this thing. We can in fact factor the portion in brackets out. = You should notice another thing as well. The Pythagorean Identity is apparent.
- 5. Elegant Problem Determine Theta Given The Equation Below: = Sin^2Θ + Cos^2Θ is equal to 1. So now we just have this. = Cos^2Θ – Cos^2Θ equal zero, so you are just left with Sin^2Θ.
- 6. Elegant Problem Determine Theta Given The Equation Below: = Now we obtain the reciprocal of the reciprocal form of sin^2Θ, which is 1/csc^2Θ. This value is equal to sin^2Θ. Cosecant is the reciprocal of Sine, so if we take the reciprocal of Cosecant, we obtain Sine. = You should notice that we can isolate 1.
- 7. Elegant Problem Determine Theta Given The Equation Below: = Hmmm… All the stuff on the left must equal 1. Maybe this will help you understand what to do. I will assign cscΘ a letter. Let Q=cscΘ = What value for Q squared times Q to the exponent Q to the exponent Q will give 1? Well, if you think about it, The only value that satisfies this equation is 1. 1^2=1 1^1=1 The values are always one, so cscΘ must equal 1.
- 9. Elegant Problem Determine Theta Given The Equation Below: = In the end, sinΘ=1. You can use your calculator for this, but you really should not have to. If you do, use the arcsine function. It is obvious that theta = 90 degrees or Π/2 radians Now let’s take a look at a unit circle to see this.
- 10. Elegant Problem
- 11. Elegant Problem Now the question did not indicate an interval for theta, so let’s just assume that it wants every angle for when sinΘ=1. Degrees Θ = 90° + K360 where K is an element of I And Radians Θ = Π/2 + K2Π where K is an element of I