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September 2021
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Geometry
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September 6, 2021
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September 6 of 2021
MODELING WITH MATHEMATICS Scientists can measure the depths of craters on the moon by looking at photos of shadows. The length of the shadow cast by the edge of a crater is 500 meters. The angle of elevation of the rays of the Sun is 55°. Estimate the depth d of the crater.
You are making a kite and need to figure out how much binding to buy. You need the binding for the perimeter of the kite. The binding comes in packages of two yards. How many packages should you buy?
In Exercises $11-14$ , Find the indicated measure. Explain your reasoning. $$ \mathrm{m} \angle \mathrm{ABD} $$
Copy and complete the two-column proof for the Congruent Supplement Theorem (Theorem 2.4). Then write a paragraph proof. (See Example 5.) $\begin{array}{ll}{\text { Given }} & {\angle 1 \text { and } \angle 2 \text { are supplementary. }} \\ {} & {\angle 3 \text { and } \angle 4 \text { are…
In Exercises $11-14,$ Ind the values of $\mathrm{x}$ and $\mathrm{y}$ . (See Example 4.)
In Exercises $13-16,$ write the negation of the statement. The dog is not a Lab.
A rhombus has diagonals of length 4 and $10 .$ Find the angles of the rhombus to the nearest degree.
$A B C D$ is a square. a. Find the distance from $H$ to each side of the square. b. Find $B F . F C, C G, D E$ . EA, EH. and HF.
In Exercises 23-34, show that $f$ and $g$ are inverse functions (a) algebraically and (b) graphically. $f(x) = \frac{x+3}{x-2}$, $g(x) = \frac{2x+3}{x-1}$
In Exercises 7-14, find the inverse function of $f$ informally. Verify that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. $f(x) = \frac{1}{3}x$
In Exercises 67-72, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.
In Exercises 43-50, evaluate the function for the indicated values. $g(x) = 2[[x]]$ (a) $g(-3)$ (b) $g(0.25)$ (c) $g(9.5)$ (d) $g(\frac{11}{3})$
In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. $ f(x) = \left\{ \begin{array}{ll} x+3, & \mbox{ $ x \le 0 $} \\ 3, & \mbox{ $ 0 < x \le 2 $} \\ 2x+1, & \mbox{ $ x > 2 $} …
In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. $f{x} = |x+1| + |x-1|$
In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. $f(x) = x^2 - 4x$
In Exercises 103-110, find the difference quotient and simplify your answer. $f(x) = x^2-x+1$, $\frac{f(2+h)-f(2)}{h}$, $h \neq 0$
thoughtful
In Exercises 65-78, find the slope-intercept form of the equation of the line passing through the points. Sketch the line. $(2, \frac{1}{2})$, $(\frac{1}{2}, \frac{5}{4})$
Match each equation of a line with its form. (a) $ Ax + By + C = 0 $ (b) $ x = a $ (c) $ y = b $ (d) $ y = mx +…
DEPRECIATION A hospital purchases a new magnetic resonance imaging (MRI) machine for $ \$500,000 $. The depreciated value $ y $ (reduced value) after $ t $ years is given by $ y = 500,000 - 40,000t, 0 \le t \le 8 $. Sketch the graph of the equation.
In Exercises 19-22, graphically estimate the $ x $- and $ y $-intercepts of the graph. Verify your results algebraically. $ y = (x-3)^2 $
In Exercises 61-64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. Original coordinates of vertices: $ (5, 8) $, $ (3, 6) $, $ (7, 6) $, $ (5, 2) $ Shift: 6 units downward, 10 units to the left
In Exercises 7-10, plot the points in the Cartesian plane. $ (3, 8) $, $ (0.5, -1) $, $ (5, -6) $, $ (-2, 2.5) $
In a circle of diameter 10 m, a regular five-pointed star touching its cricumference is inscribed. What is the area of that part not covered by the star?
MODELING WITH MATHEMATICS Scientists can measure the depths of craters on the moon by looking at photos of shadows. The length of the shadow cast by the edge of a crater is 500 meters. The angle of elevation of the rays of the Sun is 55°. Estimate the depth d of the crater.
In Exercises $11-14$ , Find the indicated measure. Explain your reasoning. $$ \mathrm{m} \angle \mathrm{ABD} $$
In Exercises $11-14,$ Ind the values of $\mathrm{x}$ and $\mathrm{y}$ . (See Example 4.)
In Exercises $13-16,$ write the negation of the statement. The dog is not a Lab.
A rhombus has diagonals of length 4 and $10 .$ Find the angles of the rhombus to the nearest degree.
In Exercises 23-34, show that $f$ and $g$ are inverse functions (a) algebraically and (b) graphically. $f(x) = \frac{x+3}{x-2}$, $g(x) = \frac{2x+3}{x-1}$
In Exercises 7-14, find the inverse function of $f$ informally. Verify that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. $f(x) = \frac{1}{3}x$
In Exercises 67-72, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.
In Exercises 43-50, evaluate the function for the indicated values. $g(x) = 2[[x]]$ (a) $g(-3)$ (b) $g(0.25)$ (c) $g(9.5)$ (d) $g(\frac{11}{3})$
In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. $ f(x) = \left\{ \begin{array}{ll} x+3, & \mbox{ $ x \le 0 $} \\ 3, & \mbox{ $ 0 < x \le 2 $} \\ 2x+1, & \mbox{ $ x > 2 $} …
In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. $f{x} = |x+1| + |x-1|$
In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. $f(x) = x^2 - 4x$
In Exercises 103-110, find the difference quotient and simplify your answer. $f(x) = x^2-x+1$, $\frac{f(2+h)-f(2)}{h}$, $h \neq 0$
thoughtful
In Exercises 65-78, find the slope-intercept form of the equation of the line passing through the points. Sketch the line. $(2, \frac{1}{2})$, $(\frac{1}{2}, \frac{5}{4})$
Match each equation of a line with its form. (a) $ Ax + By + C = 0 $ (b) $ x = a $ (c) $ y = b $ (d) $ y = mx +…
DEPRECIATION A hospital purchases a new magnetic resonance imaging (MRI) machine for $ \$500,000 $. The depreciated value $ y $ (reduced value) after $ t $ years is given by $ y = 500,000 - 40,000t, 0 \le t \le 8 $. Sketch the graph of the equation.
In Exercises 19-22, graphically estimate the $ x $- and $ y $-intercepts of the graph. Verify your results algebraically. $ y = (x-3)^2 $
In Exercises 61-64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. Original coordinates of vertices: $ (5, 8) $, $ (3, 6) $, $ (7, 6) $, $ (5, 2) $ Shift: 6 units downward, 10 units to the left
In Exercises 7-10, plot the points in the Cartesian plane. $ (3, 8) $, $ (0.5, -1) $, $ (5, -6) $, $ (-2, 2.5) $