precalc final---Here is the extraction of the content from the image: **Problem 575** **Question Stem:** 575. The figure at right shows an outermost 1 × 1 square, within which appears an inscribed circle, within which appears another inscribed square, within which appears another inscribed circle, within which appears another inscribed square. Although the figure does not show it, this process can be continued indefinitely. Let $L_1 = 1$ be the length of a side of the first (largest) square, $L_2$ be the length of a side of the second square, $L_3$ be the length of a side of the third square, and so on. Show that the numbers $L_1, L_2, L_3, \dots$ form a geometric sequence, and calculate $L_{20}$. **Chart/Diagram Description:** * **Type:** Geometric diagram illustrating nested squares and circles. * **Main Elements:** * An outermost square. * An inscribed circle within the outermost square. * An inscribed square within the circle. * An inscribed circle within the second square. * An inscribed square within the second circle. * The figure shows three squares and two circles in total, nested within each other, starting with the largest square on the outside. * The vertices of each inscribed square lie on the circumference of the circle it is inscribed within. * Each circle is tangent to the sides of the square it is inscribed within. **Problem 580** **Question Stem:** 580. Simplify by hand: **Options:** (a) $(3^{-1} + 4^{-1})^{-1}$ (b) $\frac{6^{4000}}{12^{2000}}$ (c) $7^u 7^u$ (d) $\sqrt{64x^{16}}$ (e) $2^m 3^{-m}$ **Problem 676** **Question Stem:** 676. Three squares are placed next to each other as shown. The vertices $A$, $B$, and $C$ are collinear. In terms of $m$ and $n$, find the dimension $y$. **Chart/Diagram Description:** * **Type:** Geometric diagram showing three squares placed side-by-side horizontally, with their bottom edges aligned. * **Main Elements:** * Three squares of different sizes are arranged horizontally from left to right. * The leftmost square has side length $m$. * The middle square has side length $y$. * The rightmost square has side length $n$. * The squares are placed such that their bottom edges are on a horizontal line. * Point A is the top-left vertex of the leftmost square. * Point B is the top-left vertex of the middle square. * Point C is the top-left vertex of the rightmost square. * Points A, B, and C are indicated as being collinear, forming a straight line that passes through them. **Problem 545** **Question Stem:** 545. (Continuation) The figure at right shows one of the many rectangles that can be inscribed in the ellipse $4x^2 + 9y^2 = 36$. What are the coordinates of point $(x, y)$ that correspond to the rectangle with largest area? _ [Yellow underline below the question, likely a space for an answer]_ **Equation:** $4x^2 + 9y^2 = 36$ **Chart/Diagram Description:** * **Type:** Coordinate plane with an ellipse and an inscribed rectangle. * **Main Elements:** * An ellipse centered at the origin (0,0). * Horizontal X-axis and vertical Y-axis intersecting at the origin. Both axes have tick marks. * A rectangle is drawn inscribed within the ellipse. The vertices of the rectangle appear to be on the ellipse, and its sides are parallel to the coordinate axes. * The point $(x,y)$ is labeled in the first quadrant, representing one of the vertices of the inscribed rectangle. Vertical and horizontal dashed lines extend from this point to the axes, indicating its coordinates. **Problem 572** **Question Stem:** 572. Find the $y$-intercept of the graph of $y + 1 = 2^{x-3}$. How does the graph of $y + 1 = 2^{x-3}$ compare with the graph of $y = 2^x$? How about the graph of $y = 2^{x-3} - 1$? What is the range of $y + 1 = 2^{x-3}$ and $y = 2^x$? **Equations:** $y + 1 = 2^{x-3}$ $y = 2^x$ $y = 2^{x-3} - 1$ **Problem 490** The function $F$ defined by $F(x) = 31416(1.24)^x$ describes the number of mold spores found growing on a pumpkin pie $x$ days after the mold was discovered. (a) How many spores were on the pie when the mold was first discovered? (b) How many spores were on the pie two days before the mold was discovered? (c) What is the daily rate of growth of this population? (d) What is the hourly rate of growth? (e) Let $G(x)$ be the spore count on the same pie, $x$ hours after the mold was discovered. Write a description of the function $G$. **Problem 500** Given that $\log_c 8 = 2.27$ and $\log_c 5 = 1.76$, a calculator is not needed to evaluate (a) $\log_c 40$ (b) $\log_c (5/8)$ (c) $\log_c 2$ (d) $\log_c (5^m)$ (e) $\log_c 0.04$ **Question 483** An object is moving clockwise along the elliptical path 25x² + 4y² = 100, making a complete tour every 20 seconds. The object is at (0, 5) when t = 0. (a)Write a parametric description of this motion, consistent with the given details. (b)Do your equations describe an object that is moving with a constant speed? Explain. - **Question 484** The table shown at right contains the results of an experiment in which a golf ball is dropped from the roof of the Library. For each of the displayed times, the corresponding height of the ball above the ground is given. Use this data to answer the following questions: (a)What was the average speed of the ball during the first second? (b)What was the average speed of the ball during the half-second interval from t = 0.5 to t = 1.0? (c)What was the average speed of the ball during each of the short time intervals from t = 0.9 to t = 1.0 and from t = 1.0 to t = 1.1? (d)Find an approximation for the speed of the ball when t = 1.0. (e)Find an approximation for the speed of the ball when t = 1.8. (f)Find an approximation for the speed of the ball when t = 2.2. **Table for Question 484** | t sec | h ft | |---|---| | 0.0 | 80.0 | | 0.1 | 79.8 | | 0.2 | 79.4 | | 0.3 | 78.6 | | 0.4 | 77.4 | | 0.5 | 76.0 | | 0.6 | 74.2 | | 0.7 | 72.2 | | 0.8 | 69.8 | | 0.9 | 67.0 | | 1.0 | 64.0 | | 1.1 | 60.6 | | 1.2 | 57.0 | | 1.3 | 53.0 | | 1.4 | 48.6 | | 1.5 | 44.0 | | 1.6 | 39.0 | | 1.7 | 33.8 | | 1.8 | 28.2 | | 1.9 | 22.2 | | 2.0 | 16.0 | | 2.1 | 9.4 | | 2.2 | 2.6 | **Problem 473:** On 24 August 1997 (which is the 236th day of the year), the Fidelity Select Electronics Fund reported a 44.3 percent return on investments for the year to date. Calculate the annual growth rate for this fund in 1997. Explain your method and the assumptions you made. **Problem 481:** An exponential function f is defined by f(t) = k * b^t. Its graph y = f(t) contains the points (1, 6) and (3, 24). Find the constants k and b. 463. Solve each of the following equations by hand. (a) $8^x = 32$ (b) $27^x = 243$ (c) $1000^x = 100000$ Explain why all three equations have the same solution. 470. (Continuation) Graph the curve that is described parametrically by the equation $(x, y) = (\sqrt{12} \cos t, 2 + 4 \sin t)$, for $0 \leq t \leq 360$. **Problem Number:** 462 **Question Stem:** Many ellipses are similar to $9x^2 + 25y^2 = 225$. Write an equation for the one whose focal points are: **(a)** (8,0) and (−8,0); **(b)** (0,12) and (0,−12). **Problem 446** Use a calculator to find the log of each of the following. Interpret the results. By the way, “log” is short for logarithm, to be discussed soon. (a) 1000 (b) 1000000 (c) 0.01 (d) $\sqrt{10}$ (e) $100\sqrt{10}$ (f) $\frac{1}{\sqrt[3]{100}}$ **Problem 447** Solve for x: $4^{2016} - 4^{2015} - 4^{2014} + 4^{2013} = 90 (2^x)$ **Problem 449** Using the log function, solve each of the following for x: (a) $10^x = 3$ (b) $10^x = 300$ (c) $10^x = 9$ (d) $10^x = 3^{-1}$ (e) $10^x = \sqrt{3}$ You should see a few patterns in your answers — try to explain them. **Problem 422** **Question Stem:** The figure shows a sequence of squares inscribed in the first-quadrant angle formed by the line $y = \frac{1}{2}x$ and the positive $x$-axis. Each square has two vertices on the $x$-axis and one on the line $y = \frac{1}{2}x$, and neighboring squares share a vertex. The first (smallest) square is 8 cm tall. How tall are the next four squares in the sequence? How tall is the $n^{th}$ square in the sequence? **Chart Description:** * **Type:** Geometric figure. * **Main Elements:** * **Coordinate Axes:** Positive x-axis labeled "x", positive y-axis is implied as it's the first quadrant, but not explicitly drawn or labeled as "y". The origin (0,0) is the vertex of the angle. * **Line:** A straight line labeled "$y = x/2$", starting from the origin and extending upwards into the first quadrant. * **Squares:** A sequence of five squares are drawn, decreasing in size from left to right. * Each square has its base on the x-axis. * Two vertices of each square are on the x-axis. * One vertex of each square is on the line $y = x/2$. * Neighboring squares share a common vertex that lies on the x-axis. The vertex shared is the right-bottom vertex of the left square and the left-bottom vertex of the right square. * **Angle:** The angle formed by the line $y = x/2$ and the positive x-axis. **Problem 439** **Question Stem:** 439. The cycloid. A wheel of radius 1 rolls along the $x$-axis without slipping. A mark on the rim follows a path that starts at (0,0), as shown in the figure below. (a) Find the $x$-coordinate of the point $P$ where the mark first returns to the $x$-axis. (b) Find both coordinates of the center after the wheel makes a quarter-turn. (c) Find both coordinates of the mark after the wheel makes a quarter-turn. (d) Find both coordinates of the mark after the wheel rolls a distance $t$, where $t < \frac{1}{2}\pi$. **Chart Description:** * **Type:** No chart or figure is present in the provided image for Problem 439, despite the text referring to one ("as shown in the figure below"). Here is the extraction of the content from the image: **Question 396:** Convert the following to equivalent forms in which no negative exponents appear: (a) $(\frac{2}{5})^{-1}$ (b) $\frac{6}{x^{-2}}$ (c) $(-\frac{3}{2})^{-3}$ (d) $\frac{6xy}{3x^{-1}y^{-2}}$ (e) $(\frac{2x^2}{3x^{-1}})^{-2}$ **Question 397:** Working in degree mode, find plausible equations for each of the sinusoidal graphs below: **Chart Description for Question 397:** **Chart 1:** Type: Sinusoidal line chart. Coordinate Axes: X-axis and Y-axis with grid lines. Y-axis is labeled with tic marks at -3, 0, and 3. The X-axis is labeled with tic marks and the value 180 is marked on the positive side. Graph: A smooth wave passing through the origin (0,0). It increases to a positive peak, decreases through the x-axis, reaches a negative trough, increases through the x-axis at x=180, reaches a positive peak, and continues. It appears to be a sine wave centered at y=0 with an amplitude of 3 and a period of 360 degrees (as one full cycle seems to complete roughly by x=360, suggested by the x=180 point being halfway through the second half of the first cycle). **Chart 2:** Type: Sinusoidal line chart. Coordinate Axes: X-axis and Y-axis with grid lines. Y-axis is labeled with tic marks at -3, 0, and 3. The X-axis is labeled with tic marks and the value 180 is marked on the positive side. Graph: A smooth wave starting at a positive peak on the y-axis. It decreases through the x-axis, reaches a negative trough, increases through the x-axis at x=180, reaches a positive peak, and continues. It appears to be a cosine wave centered at y=0 with an amplitude of 3 and a period of 360 degrees (as one full cycle completes by x=360, suggested by x=180 being at the trough, which is halfway through the cycle). **Chart 3:** Type: Sinusoidal line chart. Coordinate Axes: X-axis and Y-axis with grid lines. Y-axis is labeled with tic marks at -3, 0, and 3. The X-axis is labeled with tic marks and the value 120 is marked on the positive side. Graph: A smooth wave appearing to pass through or near the origin (0,0). It shows multiple cycles within a relatively short range of the x-axis compared to the other graphs. The frequency is higher, meaning the period is shorter. The value x=120 is marked. The graph appears to cross the x-axis multiple times between 0 and 120. It seems to be a sine wave centered at y=0 with an amplitude of 3. The period seems to be 120 degrees, as a full cycle appears to complete around x=120. **Question 413:** Convert the following to simpler equivalent forms: (a) $x^6x^{-6}$ (b) $(8a^{-3}b^6)^{1/3}$ (c) $(\frac{x^{1/2}}{y^{2/3}})^6 (\frac{x^{1/2}}{y^{2/3}})^{-6}$ **Question 362** Given that tan θ = 2.4, with 180 < θ < 270, without a calculator, find the exact values of sin θ and cos θ. Are your answers rational numbers? **Question 384** Solve for x by hand. (a) x⁵ = a³ (b) x¹/⁵ = a³ (c) (1 + x)¹⁵.⁶ = 2.0 (d) x⁻² = a **Question 385** In a circle of radius 5 cm, how long is a 1-radian arc? How long is a 2.2-radian arc? **Question 333:** For each of the following, there are two points on the unit circle that fit the given description. Without finding $\theta$, describe how the two points are related to each other. (a) $\cos \theta = -0.4540$ (b) $\sin \theta = 0.6820$ (c) $\tan \theta = -1.280$ **Question 357:** Find equivalent ways to rewrite (without using a calculator) the following expressions: (a) $\frac{6a^8}{3a^4}$ (b) $(3p^3 q^4)^2$ (c) $b^{1/2} b^{1/3} b^{1/6}$ (d) $(\frac{2x^3}{3y^2})^2$ (e) $(d^{1/2})^6$ **Question 362:** Given that $\tan \theta = 2.4$, with $180 < \theta < 270$, without a calculator, find the exact values of $\sin \theta$ and $\cos \theta$. Are your answers rational numbers? **Problem 300** **Question Stem:** 300. For each of the following pairs of matrices, calculate MN and NM: **(a)** M = $\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$ and N = $\begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}$ **(b)** M = $\begin{bmatrix} 3 & -4 \end{bmatrix}$ and N = $\begin{bmatrix} 12 \\ 5 \end{bmatrix}$ **(c)** M = $\begin{bmatrix} 1 & 2 \\ -6 & -3 \end{bmatrix}$ and N = $\begin{bmatrix} -4 & 1 \\ -3 & -6 \end{bmatrix}$ --- **Problem 320** **Question Stem:** 320. Convert the polar pair $(r = 8; \theta = 150)$ to an equivalent Cartesian pair $(x, y)$. **Problem 205** **Question Stem:** 205. Revisit the 1-foot radius circular saw blade with the one red tooth. Look at the ratio m of the height h to the horizontal displacement p, that is m = h/p. The red tooth starts at the rightmost point of the saw and rotates at one degree per second counterclockwise. What is m after 37 seconds? After 137 seconds? After 237 seconds? After t seconds? Draw by hand a graph that shows how m is a function of the elapsed time t. What does the ratio m = h/p tell you about the line OT from the saw center to the tooth? **Chart Description:** * Type: Diagram of a circle representing a saw blade with a point T on its circumference. * Main Elements: * Circle: Represents the saw blade. It has a toothed edge. * Center: Labeled O, located at the origin of a coordinate system. * Point T: A point on the circumference, representing the red tooth. * Lines: * A horizontal line passing through O, labeled "table" on the right. * A vertical line passing through O. * A line segment from O to T. * A dashed vertical line segment from T down to the horizontal line through O. The length of this segment is labeled 'h'. * A dashed horizontal line segment from T to the vertical line through O. The length of this segment is labeled 'p'. * Labels: O (center), T (tooth), h (height), p (horizontal displacement), "table" (horizontal reference). * Relative Position: O is at the center. T is on the circumference. h is the vertical distance from T to the horizontal line through O. p is the horizontal distance from T to the vertical line through O. * Note: The lines labeled h and p form a right triangle with the line segment OT, where OT is the hypotenuse. The radius of the saw blade is given as 1 foot in the question stem. **Problem 253** **Question Stem:** 253. Three tennis balls fit snugly inside a cylindrical can. What percent of the available space inside the can is occupied by the balls? **Chart Description:** * Type: Diagram showing a cylinder containing three spheres. * Main Elements: * Cylinder: Represented by an oval for the front circular face and a horizontal rectangle shape for the body, with dashed lines for the back circular face. * Spheres: Three circles representing the tennis balls, arranged in a single row inside the cylinder, touching each other and the walls of the cylinder. The circles are shown with dashed lines where they are behind the front surface of the cylinder. * Relative Position: The three spheres are positioned along the length of the cylinder, fitting "snugly," implying they touch the top, bottom, and sides of the cylinder as well as each other. **Question 108** **Question Stem:** 108. The figure at right shows a long rectangular strip of paper, one corner of which has been folded over to meet the opposite edge, thereby creating a 30-degree angle. Given that the width of the strip is 12 inches, find the length of the crease. **Chart/Diagram Description:** * **Type:** Geometric figure illustrating a folded rectangular strip of paper. * **Main Elements:** * A horizontal rectangular strip representing the paper. * A portion of the top-right corner of the strip is shown as folded downwards and to the left. * The folded part forms a triangle. * The crease is a line segment forming the base of this triangle. * A dashed line indicates the original top edge of the strip. * An angle of 30 degrees is labeled within the folded triangular region, near the original top edge. * A vertical double-headed arrow indicates the width of the rectangular strip. * The width is labeled as 12 inches (12"). * The folded corner point meets a point on the bottom edge of the original strip (or the opposite edge as described in the text). **Question 118** **Question Stem:** 118. Without a calculator, find all solutions w between 0 and 360, inclusive, providing diagrams that support your results. **Options:** (a) cos w = cos(−340) (b) cos w = sin 20 (c) sin w = cos(−10) (d) sin w < - 1/2 (e) 1 < tan w