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Questions in mathematics

📝 Answered - Give the slope and the $y$-intercept of the line with the given equation. $y=3 x+7$ What is the slope? Select the correct choice below and fill in any answer boxes within your choice. A. The slope is 3. B. The slope is undefined. What is the y-intercept? Select the correct choice below and fill in any answer boxes within your choice. A. The y -intercept is 7. B. There is no $y$-intercept. Graph the equation.

📝 Answered - Consider the relationship below, given [tex]$\frac{\pi}{2}\ \textless \ \theta\ \textless \ \pi$[/tex]. [tex]$\sin ^2 \theta+\cos ^2 \theta=1$[/tex] Which of the following best explains how this relationship and the value of [tex]$\sin \theta$[/tex] can be used to find the other trigonometric values? A. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for [tex]$\cos \theta$[/tex] finds the unknown leg, and then all other trigonometric values can be found. B. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the angles of a right triangle; therefore, solving the relationship will find all three angles of the triangle, and then all trigonometric values can be found. C. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the angles of a right triangle; therefore, other pairs of trigonometric ratios will have the same sum, 1, which can then be used to find all other values. D. The values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] represent the legs of a right triangle with a hypotenuse of -1, since [tex]$\theta$[/tex] is in Quadrant II; therefore, solving for [tex]$\cos \theta$[/tex] finds the unknown leg, and then all other trigonometric values can be found.

📝 Answered - Cone W has a radius of 6 cm and a height of 5 cm. Square pyramid X has the same base area and height as cone W. Paul and Manuel disagree on the reason why the volumes of cone W and square pyramid X are related. | Paul | Manuel | |---|---| | The volume of square pyramid X is equal to the volume of cone W. This can be proven by finding the base area and volume of cone [tex]$W$[/tex], along with the volume of square pyramid X. | The volume of square pyramid X is equal to the volume of cone W. This can be proven by finding the base area and volume of cone [tex]$W$[/tex], along with the volume of square pyramid X. | | The base area of cone [tex]$W$[/tex] is [tex]$n(d)=n(12)=37.68 cm^2$[/tex]. | The base area of cone [tex]$W$[/tex] is [tex]$\pi(r^2)=\pi(6^2)=113.04 cm^2$[/tex]. | | The volume of cone W is [tex]$\frac{1}{3}[/tex] (area of base)( h )=[tex]$\frac{1}{3}(37.68)(5)=62.8$[/tex] [tex]$cm ^3$[/tex]. | The volume of cone [tex]$W$[/tex] is [tex]$\frac{1}{3}[/tex] (area of base)(h) =[tex]$\frac{1}{3}(113.04)(5)=[/tex] [tex]$188.4 cm^3$[/tex]. | | The volume of square pyramid X is [tex]$\frac{1}{3}[/tex] (area of base)(h) =[tex]$\frac{1}{3}(37.68)$[/tex] (5) [tex]$=62.8 cm^3$[/tex]. | The volume of square pyramid [tex]$X$[/tex] is [tex]$\frac{1}{3}[/tex] (area of base)(h) =[tex]$\frac{1}{3}$[/tex] [tex]$(113.04)(5)=188.4 cm^3$[/tex]. | Examine their arguments. Which statement explains whose argument is correct and why? A. Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid X. B. Paul's argument is correct; Manuel used the incorrect base area to find the volumes of square pyramid X and cone W. C. Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid X. D. Manuel's argument is correct; Paul used the incorrect base area to find the volumes of square pyramid X and cone W.

📝 Answered - Solve the inequality: [tex]9-4 r\ \textgreater \ 5[/tex] A. [tex]r\ \textgreater \ 1[/tex] B. [tex]r\ \textless \ -4[/tex] C. [tex]r\ \textless \ 1[/tex] D. [tex]r\ \textless \ -1[/tex]

📝 Answered - Select the correct answer. A tree has 99,400 leaves before autumn. Once autumn begins, the number of leaves on a tree decreases at a rate of $13 \%$ per day. After $t$ days of autumn, there are fewer than 12,000 leaves on the tree. Which inequality represents this situation, and after how many days of autumn will the number of leaves on the tree be fewer than [tex]12,000[/tex]? A. [tex]99,400(1.13)^t\ \textgreater \ 12,000 ; 15[/tex] days B. [tex]99,400(0.87)^t\ \textless \ 12,000 ; 16[/tex] days C. [tex]12,000(1.087)^t\ \textgreater \ 99,400 ; 18[/tex] days D. [tex]12,000(0.913)^t\ \textless \ 99,400 ; 17[/tex] days

📝 Answered - $f(x)=2 x^3-8 x^2-2 x+5$

📝 Answered - The equation of a circle is $(x-3)^2+y^2=18$. The line with equation $y=m x+c$ passes through the point $(0,-9)$ and is a tangent to the circle. Find the two possible values of $m$ and, for each value of $m$, find the coordinates of the point at which the tangent touches the circle. [8]

📝 Answered - A sector is cut from a circle of radius 21 cm. The angle of the sector is 150 degrees. Find the length of the arc. Find the perimeter of the sector (take π).

📝 Answered - A treasure map says that a treasure is buried so that it partitions the distance between a rock and a tree in a 5:9 ratio. Marina traced the map onto a coordinate plane to find the exact location of the treasure. [tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex] [tex]v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1[/tex] What are the coordinates of the treasure? If necessary, round the coordinates to the nearest tenth. A. (11.4, 14.2) B. (7.6, 8.8) C. (5.7, 7.5) D. (10.2, 12.6)

📝 Answered - Find the arc length of the curve defined by the parametric equations [tex]$x(t)=-2 e^{-2 t}$[/tex] and [tex]$y(t)=3 e^{-3 t}$[/tex] from [tex]$0.6 \leq t \leq 1.2$[/tex]. Round your answer to the nearest hundredth. Provide your answer below: [tex]$s \approx$[/tex] $\square$