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college_math.PRECALCULUS
|
exercise.6.2.20
|
Use the properties of logarithms to write the expression as a single logarithm: $2 \ln (x)-3 \ln (y)-4 \ln (z)$
|
$\ln \left(\frac{x^{2}}{y^{3} z^{4}}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\ln \left(\frac{x^{2}}{y^{3} z^{4}}\right)
|
college_math.PRECALCULUS
|
exercise.8.2.25
|
Solve the following system of linear equations: $\left\{\begin{aligned} 2 x-3 y+z & =-1 \\ 4 x-4 y+4 z & =-13 \\ 6 x-5 y+7 z & =-25\end{aligned}\right.$
|
$\left(-2 t-\frac{35}{4},-t-\frac{11}{2}, t\right)$ for all real numbers $t$
|
Creative Commons License
|
college_math.precalculus
|
\left(-2 t-\frac{35}{4},-t-\frac{11}{2}, t\right)$ for all real numbers $t
|
college_math.PRECALCULUS
|
exercise.10.2.49
|
Approximate the given value to three decimal places: $\sin \left(78.95^{\circ}\right)$
|
$\sin \left(78.95^{\circ}\right) \approx 0.981$
|
Creative Commons License
|
college_math.precalculus
|
\sin \left(78.95^{\circ}\right) \approx 0.981
|
college_math.PRECALCULUS
|
exercise.6.3.8
|
Solve the equation analytically: $9 \cdot 3^{7 x}=\left(\frac{1}{9}\right)^{2 x}$
|
$x=-\frac{2}{11}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{2}{11}
|
college_math.PRECALCULUS
|
exercise.10.7.93
|
Solve the given inequality: $\arcsin (2 x)>0$
|
$\left(0, \frac{1}{2}\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left(0, \frac{1}{2}\right]
|
college_math.PRECALCULUS
|
exercise.6.3.44
|
Use your calculator to help you solve the inequality: $3^{(x-1)}<2^{x}$
|
$\approx(-\infty, 2.7095)$
|
Creative Commons License
|
college_math.precalculus
|
\approx(-\infty, 2.7095)
|
college_math.PRECALCULUS
|
exercise.10.7.3
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\sin (-2 x)=\frac{\sqrt{3}}{2}$
|
$x=\frac{2 \pi}{3}+\pi k$ or $x=\frac{5 \pi}{6}+\pi k ; x=\frac{2 \pi}{3}, \frac{5 \pi}{6}, \frac{5 \pi}{3}, \frac{11 \pi}{6}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{2 \pi}{3}+\pi k$ or $x=\frac{5 \pi}{6}+\pi k ; x=\frac{2 \pi}{3}, \frac{5 \pi}{6}, \frac{5 \pi}{3}, \frac{11 \pi}{6}
|
college_math.PRECALCULUS
|
exercise.3.4.49
|
Create a polynomial $f$ with real number coefficients that has the following characteristics:
- The zeros of $f$ are $c=\pm 1$ and $c=\pm i$
- The leading term of $f(x)$ is $42x^4$
|
$f(x)=42(x-1)(x+1)(x-i)(x+i)$
|
Creative Commons License
|
college_math.precalculus
|
f(x)=42(x-1)(x+1)(x-i)(x+i)
|
college_math.PRECALCULUS
|
exercise.6.3.6
|
Solve the equation analytically: $2^{\left(x^{3}-x\right)}=1$
|
$x=-1,0,1$
|
Creative Commons License
|
college_math.precalculus
|
x=-1,0,1
|
college_math.PRECALCULUS
|
exercise.8.2.20
|
Solve the following system of linear equations: $\left\{\begin{aligned} 2 x-4 y+z & =-7 \\ x-2 y+2 z & =-2 \\ -x+4 y-2 z & =3\end{aligned}\right.$
|
$\left(-3, \frac{1}{2}, 1\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(-3, \frac{1}{2}, 1\right)
|
college_math.PRECALCULUS
|
exercise.10.7.54
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (4 x)=\cos (2 x)$
|
$x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.4.3.7
|
Solve the rational inequality and express your answer using interval notation: $\frac{1}{x+2} \geq 0$
|
$(-2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-2, \infty)
|
college_math.PRECALCULUS
|
exercise.10.1.36
|
Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $-225^{\circ}$
|
$-\frac{5 \pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
-\frac{5 \pi}{4}
|
college_math.PRECALCULUS
|
exercise.2.2.21
|
Solve the equation: $3|x-1|=2|x+1|$
|
$x=\frac{1}{5}$ or $x=5$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{1}{5}$ or $x=5
|
college_math.PRECALCULUS
|
exercise.10.2.62
|
If $\theta=5^{\circ}$ and the hypotenuse has length 10 , how long is the side opposite $\theta$ ?
|
The side opposite $\theta$ has length $10 \sin \left(5^{\circ}\right) \approx 0.872$.
|
Creative Commons License
|
college_math.precalculus
|
The side opposite $\theta$ has length $10 \sin \left(5^{\circ}\right) \approx 0.872$.
|
college_math.PRECALCULUS
|
exercise.6.2.12
|
Expand the given logarithm and simplify: $\log _{6}\left(\frac{216}{x^{3} y}\right)^{4}$
|
$12-12 \log _{6}(x)-4 \log _{6}(y)$
|
Creative Commons License
|
college_math.precalculus
|
12-12 \log _{6}(x)-4 \log _{6}(y)
|
college_math.PRECALCULUS
|
exercise.2.2.15
|
Solve the equation: $\left|x^{2}-1\right|=3$
|
$x=-2$ or $x=2$
|
Creative Commons License
|
college_math.precalculus
|
x=-2$ or $x=2
|
college_math.PRECALCULUS
|
exercise.6.4.26
|
Solve the inequality analytically: $x \ln (x)-x>0$
|
$(e, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(e, \infty)
|
college_math.PRECALCULUS
|
exercise.6.1.47
|
Find the domain of the function: $f(x)=\log \left(\frac{x+2}{x^{2}-1}\right)$
|
$(-2,-1) \cup(1, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-2,-1) \cup(1, \infty)
|
college_math.PRECALCULUS
|
exercise.6.3.27
|
Solve the equation analytically: $7^{3+7 x}=3^{4-2 x}$
|
$x=\frac{4 \ln (3)-3 \ln (7)}{7 \ln (7)+2 \ln (3)}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{4 \ln (3)-3 \ln (7)}{7 \ln (7)+2 \ln (3)}
|
college_math.PRECALCULUS
|
exercise.1.1.9
|
Write the set using interval notation: $\{x \mid x \neq-1\}$
|
$(-\infty,-1) \cup(-1, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-1) \cup(-1, \infty)
|
college_math.PRECALCULUS
|
exercise.11.9.21
|
Find the work done pushing a 200 pound barrel 10 feet up a $12.5^{\circ}$ incline. Ignore all forces acting on the barrel except gravity, which acts downwards. Round your answer to two decimal places.
HINT: Since you are working to overcome gravity only, the force being applied acts directly upwards. This means that the angle between the applied force in this case and the motion of the object is not the $12.5^{\circ}$ of the incline!
|
(1500 pounds) $\left(300\right.$ feet) $\cos \left(0^{\circ}\right)=450,000$ foot-pounds
|
Creative Commons License
|
college_math.precalculus
|
(1500 pounds) $\left(300\right.$ feet) $\cos \left(0^{\circ}\right)=450,000$ foot-pounds
|
college_math.PRECALCULUS
|
exercise.9.1.2
|
Write out the first four terms of the given sequence: $d_{j}=(-1)^{\frac{j(j+1)}{2}}, j \geq 1$
|
$-1,-1,1,1$
|
Creative Commons License
|
college_math.precalculus
|
-1,-1,1,1
|
college_math.PRECALCULUS
|
exercise.11.4.66
|
Convert the equation from polar coordinates into rectangular coordinates: $r=3 \sin (\theta)$
|
$r=\frac{19}{4 \cos (\theta)-\sin (\theta)}$
|
Creative Commons License
|
college_math.precalculus
|
r=\frac{19}{4 \cos (\theta)-\sin (\theta)}
|
college_math.PRECALCULUS
|
exercise.10.7.52
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $3 \sqrt{3} \sin (3 x)-3 \cos (3 x)=3 \sqrt{3}$
|
$x=\frac{\pi}{6}, \frac{5 \pi}{18}, \frac{5 \pi}{6}, \frac{17 \pi}{18}, \frac{3 \pi}{2}, \frac{29 \pi}{18}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{6}, \frac{5 \pi}{18}, \frac{5 \pi}{6}, \frac{17 \pi}{18}, \frac{3 \pi}{2}, \frac{29 \pi}{18}
|
college_math.PRECALCULUS
|
exercise.10.1.39
|
Convert the angle from radian measure into degree measure: $\frac{7 \pi}{6}$
|
$210^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
210^{\circ}
|
college_math.PRECALCULUS
|
exercise.10.7.76
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $2 \cos (x) \geq 1$
|
$\left[0, \frac{\pi}{2}\right) \cup\left(\frac{11 \pi}{6}, 2 \pi\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left[0, \frac{\pi}{2}\right) \cup\left(\frac{11 \pi}{6}, 2 \pi\right]
|
college_math.PRECALCULUS
|
exercise.4.3.24
|
A faucet can fill a sink in 5 minutes while a drain will empty the same sink in 8 minutes. If the faucet is turned on and the drain is left open, how long will it take to fill the sink?
|
$\frac{40}{3} \approx 13.33$ minutes
|
Creative Commons License
|
college_math.precalculus
|
\frac{40}{3} \approx 13.33$ minutes
|
college_math.PRECALCULUS
|
exercise.7.3.21
|
A parabolic arch is constructed which is 6 feet wide at the base and 9 feet tall in the middle. Find the height of the arch exactly 1 foot in from the base of the arch.
|
The arch can be modeled by $x^{2}=-(y-9)$ or $y=9-x^{2}$. One foot in from the base of the arch corresponds to either $x= \pm 2$, so the height is $y=9-( \pm 2)^{2}=5$ feet.
|
Creative Commons License
|
college_math.precalculus
|
The arch can be modeled by $x^{2}=-(y-9)$ or $y=9-x^{2}$. One foot in from the base of the arch corresponds to either $x= \pm 2$, so the height is $y=9-( \pm 2)^{2}=5$ feet.
|
college_math.PRECALCULUS
|
exercise.10.7.38
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos ^{3}(x)=-\cos (x)$
|
$x=\frac{\pi}{2}, \frac{3 \pi}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{2}, \frac{3 \pi}{2}
|
college_math.PRECALCULUS
|
exercise.6.2.6
|
Expand the given logarithm and simplify: $\log _{5}\left(x^{2}-25\right)$
|
$\log _{5}(x-5)+\log _{5}(x+5)$
|
Creative Commons License
|
college_math.precalculus
|
\log _{5}(x-5)+\log _{5}(x+5)
|
college_math.PRECALCULUS
|
exercise.2.4.15
|
The International Silver Strings Submarine Band holds a bake sale each year to fund their trip to the National Sasquatch Convention. It has been determined that the cost in dollars of baking $x$ cookies is $C(x)=0.1 x+25$ and that the demand function for their cookies is $p=10-.01 x$. How many cookies should they bake in order to maximize their profit?
|
$\left(1, \frac{5}{3}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(1, \frac{5}{3}\right)
|
college_math.PRECALCULUS
|
exercise.6.1.28
|
Evaluate the expression: $\log _{4}(8)$
|
$\log _{4}(8)=\frac{3}{2}$
|
Creative Commons License
|
college_math.precalculus
|
\log _{4}(8)=\frac{3}{2}
|
college_math.PRECALCULUS
|
exercise.10.1.40
|
Convert the angle from radian measure into degree measure: $\frac{11 \pi}{6}$
|
$330^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
330^{\circ}
|
college_math.PRECALCULUS
|
exercise.6.1.49
|
Find the domain of the function: $f(x)=\ln (7-x)+\ln (x-4)$
|
$(4,7)$
|
Creative Commons License
|
college_math.precalculus
|
(4,7)
|
college_math.PRECALCULUS
|
exercise.6.3.40
|
Use your calculator to help you solve the equation: $2^{x}=x^{2}$
|
$x \approx-0.76666, x=2, x=4$
|
Creative Commons License
|
college_math.precalculus
|
x \approx-0.76666, x=2, x=4
|
college_math.PRECALCULUS
|
exercise.1.1.13
|
Write the set using interval notation: $\{x \mid x \neq 0, \pm 4\}$
|
$(-\infty,-4) \cup(-4,0) \cup(0,4) \cup(4, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-4) \cup(-4,0) \cup(0,4) \cup(4, \infty)
|
college_math.PRECALCULUS
|
exercise.3.3.36
|
Find the real solutions of the polynomial equation $9 x^{2}+5 x^{3}=6 x^{4}$.
|
$x=0, \frac{5 \pm \sqrt{241}}{12}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, \frac{5 \pm \sqrt{241}}{12}
|
college_math.PRECALCULUS
|
exercise.10.7.30
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cot ^{2}(x)=3 \csc (x)-3$
|
$x=\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{6}, \frac{11 \pi}{6}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{6}, \frac{11 \pi}{6}
|
college_math.PRECALCULUS
|
exercise.9.1.7
|
Write out the first four terms of the given sequence: $a_{1}=3, a_{n+1}=a_{n}-1, n \geq 1$
|
$3,2,1,0$
|
Creative Commons License
|
college_math.precalculus
|
3,2,1,0
|
college_math.PRECALCULUS
|
exercise.2.1.34
|
The Topology Taxi Company charges $\$ 2.50$ for the first fifth of a mile and $\$ 0.45$ for each additional fifth of a mile. Find a linear function which models the taxi fare $F$ as a function of the number of miles driven, $m$. Interpret the slope of the linear function and find and interpret $F(0)$.
|
$F(m)=2.25 m+2.05$ The slope 2.25 means it costs an additional $\$ 2.25$ for each mile beyond the first 0.2 miles. $F(0)=2.05$, so according to the model, it would cost $\$ 2.05$ for a trip of 0 miles. Would this ever really happen? Depends on the driver and the passenger, we suppose.
|
Creative Commons License
|
college_math.precalculus
|
F(m)=2.25 m+2.05$ The slope 2.25 means it costs an additional $\$ 2.25$ for each mile beyond the first 0.2 miles. $F(0)=2.05$, so according to the model, it would cost $\$ 2.05$ for a trip of 0 miles. Would this ever really happen? Depends on the driver and the passenger, we suppose.
|
college_math.PRECALCULUS
|
exercise.6.3.38
|
Solve the inequality analytically: $\frac{150}{1+29 e^{-0.8 t}} \leq 130$
|
$\left(-\infty, \frac{\ln \left(\frac{2}{377}\right)}{-0.8}\right]=\left(-\infty, \frac{5}{4} \ln \left(\frac{377}{2}\right)\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\infty, \frac{\ln \left(\frac{2}{377}\right)}{-0.8}\right]=\left(-\infty, \frac{5}{4} \ln \left(\frac{377}{2}\right)\right]
|
college_math.PRECALCULUS
|
exercise.10.7.97
|
Solve the given inequality: $2 \arcsin (x)^{2}>\pi \arcsin (x)$
|
$[-1,0)$
|
Creative Commons License
|
college_math.precalculus
|
[-1,0)
|
college_math.PRECALCULUS
|
exercise.10.7.99
|
Express the domain of the function using the extended interval notation: $f(x)=\frac{1}{\cos (x)-1}$
|
$\bigcup_{k=-\infty}^{\infty}(2 k \pi,(2 k+2) \pi)$
|
Creative Commons License
|
college_math.precalculus
|
\bigcup_{k=-\infty}^{\infty}(2 k \pi,(2 k+2) \pi)
|
college_math.PRECALCULUS
|
exercise.9.4.7
|
Evaluate: $\left(\begin{array}{l}8 \\ 3\end{array}\right)$
|
56
|
Creative Commons License
|
college_math.precalculus
|
56
|
college_math.PRECALCULUS
|
exercise.3.4.21
|
Simplify the given power of $i$: $i^7$
|
$i^{7}=i^{4} \cdot i^{3}=1 \cdot(-i)=-i$
|
Creative Commons License
|
college_math.precalculus
|
i^{7}=i^{4} \cdot i^{3}=1 \cdot(-i)=-i
|
college_math.PRECALCULUS
|
exercise.11.4.50
|
Convert the equation from rectangular coordinates into polar coordinates: $x^{2}+y^{2}-2 y=0$
|
$(20, \pi-\arctan (3))$
|
Creative Commons License
|
college_math.precalculus
|
(20, \pi-\arctan (3))
|
college_math.PRECALCULUS
|
exercise.9.2.11
|
Rewrite the sum using summation notation: $x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}$
|
$\sum_{k=1}^{4}(-1)^{k-1} \frac{x^{2 k-1}}{2 k-1}$
|
Creative Commons License
|
college_math.precalculus
|
\sum_{k=1}^{4}(-1)^{k-1} \frac{x^{2 k-1}}{2 k-1}
|
college_math.PRECALCULUS
|
exercise.6.2.27
|
Use the properties of logarithms to write the expression as a single logarithm: $\log _{2}(x)+\log _{4}(x)$
|
$\log _{2}\left(x^{3 / 2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\log _{2}\left(x^{3 / 2}\right)
|
college_math.PRECALCULUS
|
exercise.8.5.18
|
Find the inverse of the given matrix: $F=\left[\begin{array}{rrr}4 & 6 & -3 \\ 3 & 4 & -3 \\ 1 & 2 & 6\end{array}\right]$
|
$F^{-1}=\left[\begin{array}{rrr}-\frac{5}{2} & \frac{7}{2} & \frac{1}{2} \\ \frac{7}{4} & -\frac{9}{4} & -\frac{1}{4} \\ -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right]$
|
Creative Commons License
|
college_math.precalculus
|
F^{-1}=\left[\begin{array}{rrr}-\frac{5}{2} & \frac{7}{2} & \frac{1}{2} \\ \frac{7}{4} & -\frac{9}{4} & -\frac{1}{4} \\ -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right]
|
college_math.PRECALCULUS
|
exercise.4.3.9
|
Solve the rational inequality and express your answer using interval notation: $\frac{x}{x^{2}-1}>0$
|
$(-1,0) \cup(1, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-1,0) \cup(1, \infty)
|
college_math.PRECALCULUS
|
exercise.10.2.43
|
Solve the equation for $t$: $\sin (t)=-\frac{1}{2}$
|
$\sin (t)=-\frac{1}{2}$ when $t=\frac{7 \pi}{6}+2 \pi k$ or $t=\frac{11 \pi}{6}+2 \pi k$ for any integer $k$.
|
Creative Commons License
|
college_math.precalculus
|
\sin (t)=-\frac{1}{2}$ when $t=\frac{7 \pi}{6}+2 \pi k$ or $t=\frac{11 \pi}{6}+2 \pi k$ for any integer $k$.
|
college_math.PRECALCULUS
|
exercise.10.1.38
|
Convert the angle from radian measure into degree measure: $-\frac{2 \pi}{3}$
|
$-120^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
-120^{\circ}
|
college_math.PRECALCULUS
|
exercise.6.3.33
|
Solve the equation analytically: $3^{x}+25 \cdot 3^{-x}=10$
|
$x=\frac{\ln (5)}{\ln (3)}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\ln (5)}{\ln (3)}
|
college_math.PRECALCULUS
|
exercise.1.3.5
|
Determine whether or not the relation represents $y$ as a function of $x$ and find the domain and range of those relations which are functions:
$\{(x, y) \mid x$ is an odd integer, and $y$ is an even integer $\}$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.6.2.8
|
Expand the given logarithm and simplify: $\log _{\frac{1}{3}}\left(9 x\left(y^{3}-8\right)\right)$
|
$-2+\log _{\frac{1}{3}}(x)+\log _{\frac{1}{3}}(y-2)+\log _{\frac{1}{3}}\left(y^{2}+2 y+4\right)$
|
Creative Commons License
|
college_math.precalculus
|
-2+\log _{\frac{1}{3}}(x)+\log _{\frac{1}{3}}(y-2)+\log _{\frac{1}{3}}\left(y^{2}+2 y+4\right)
|
college_math.PRECALCULUS
|
exercise.8.7.16
|
A certain bacteria culture follows the Law of Uninbited Growth, Equation 6.4. After 10 minutes, there are 10,000 bacteria. Five minutes later, there are 14,000 bacteria. How many bacteria were present initially? How long before there are 50,000 bacteria?
|
Initially, there are $\frac{250000}{49} \approx 5102$ bacteria. It will take $\frac{5 \ln (49 / 5)}{\ln (7 / 5)} \approx 33.92$ minutes for the colony to grow to 50,000 bacteria.
|
Creative Commons License
|
college_math.precalculus
|
Initially, there are $\frac{250000}{49} \approx 5102$ bacteria. It will take $\frac{5 \ln (49 / 5)}{\ln (7 / 5)} \approx 33.92$ minutes for the colony to grow to 50,000 bacteria.
|
college_math.PRECALCULUS
|
exercise.2.2.11
|
Solve the equation: $4-|x|=2 x+1$
|
$x=1$
|
Creative Commons License
|
college_math.precalculus
|
x=1
|
college_math.PRECALCULUS
|
exercise.2.3.15
|
The International Silver Strings Submarine Band holds a bake sale each year to fund their trip to the National Sasquatch Convention. It has been determined that the cost in dollars of baking $x$ cookies is $C(x)=0.1 x+25$ and that the demand function for their cookies is $p=10-.01 x$. How many cookies should they bake in order to maximize their profit?
|
495 cookies
|
Creative Commons License
|
college_math.precalculus
|
495 cookies
|
college_math.PRECALCULUS
|
exercise.10.2.11
|
Find the exact value of the cosine and sine of the given angle: $\theta=\frac{3 \pi}{2}$
|
$\cos \left(\frac{3 \pi}{2}\right)=0, \sin \left(\frac{3 \pi}{2}\right)=-1$
|
Creative Commons License
|
college_math.precalculus
|
\cos \left(\frac{3 \pi}{2}\right)=0, \sin \left(\frac{3 \pi}{2}\right)=-1
|
college_math.PRECALCULUS
|
exercise.8.2.10
|
Solve the following system of linear equations: $\left\{\begin{aligned} x-2 y+3 z & =7 \\ -3 x+y+2 z & =-5 \\ 2 x+2 y+z & =3\end{aligned}\right.$
|
Inconsistent
|
Creative Commons License
|
college_math.precalculus
|
Inconsistent
|
college_math.PRECALCULUS
|
exercise.10.7.87
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\csc (x)>1$
|
$\left(-2 \pi,-\frac{3 \pi}{2}\right) \cup\left(-\frac{3 \pi}{2},-\pi\right) \cup\left(0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(-2 \pi,-\frac{3 \pi}{2}\right) \cup\left(-\frac{3 \pi}{2},-\pi\right) \cup\left(0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right)
|
college_math.PRECALCULUS
|
exercise.10.7.106
|
Express the domain of the function using the extended interval notation: $f(x)=\ln (|\cos (x)|)$
|
$\bigcup_{k=-\infty}^{\infty}\left(\frac{(2 k-1) \pi}{2}, \frac{(2 k+1) \pi}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\bigcup_{k=-\infty}^{\infty}\left(\frac{(2 k-1) \pi}{2}, \frac{(2 k+1) \pi}{2}\right)
|
college_math.PRECALCULUS
|
exercise.3.4.14
|
Simplify the quantity $\sqrt{(-25)(-4)}$
|
10
|
Creative Commons License
|
college_math.precalculus
|
10
|
college_math.PRECALCULUS
|
exercise.9.2.31
|
Express the repeating decimal as a fraction of integers: $10 . \overline{159}$
|
$\frac{3383}{333}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{3383}{333}
|
college_math.PRECALCULUS
|
exercise.11.4.56
|
Convert the equation from rectangular coordinates into polar coordinates: $4 x^{2}+4\left(y-\frac{1}{2}\right)^{2}=1$
|
$(\sqrt{13}, \pi-\arctan (2))$
|
Creative Commons License
|
college_math.precalculus
|
(\sqrt{13}, \pi-\arctan (2))
|
college_math.PRECALCULUS
|
exercise.3.4.15
|
Simplify the quantity $\sqrt{-9}\sqrt{-16}$
|
-12
|
Creative Commons License
|
college_math.precalculus
|
-12
|
college_math.PRECALCULUS
|
exercise.6.2.4
|
Expand the given logarithm and simplify: $\log \left(1.23 \times 10^{37}\right)$
|
$\log (1.23)+37$
|
Creative Commons License
|
college_math.precalculus
|
\log (1.23)+37
|
college_math.PRECALCULUS
|
exercise.10.7.36
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan ^{3}(x)=3 \tan (x)$
|
$x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.2.4.17
|
The temperature $T$, in degrees Fahrenheit, $t$ hours after $6 \mathrm{AM}$ is given by $T(t)=-\frac{1}{2} t^{2}+8 t+32, \quad 0 \leq t \leq 12$. What is the warmest temperature of the day? When does this happen?
|
$(-\infty,-3] \cup[1, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-3] \cup[1, \infty)
|
college_math.PRECALCULUS
|
exercise.3.3.40
|
Find the real solutions of the polynomial equation $2 x^{3}=19 x^{2}-49 x+20$.
|
$x=\frac{1}{2}, 4,5$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{1}{2}, 4,5
|
college_math.PRECALCULUS
|
exercise.6.2.3
|
Expand the given logarithm and simplify: $\log _{5}\left(\frac{z}{25}\right)^{3}$
|
$3 \log _{5}(z)-6$
|
Creative Commons License
|
college_math.precalculus
|
3 \log _{5}(z)-6
|
college_math.PRECALCULUS
|
exercise.10.7.28
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $2 \sec ^{2}(x)=3-\tan (x)$
|
$x=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{\pi}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{\pi}{2}
|
college_math.PRECALCULUS
|
exercise.8.7.19
|
Solve the system of nonlinear equations after making the appropriate substitutions: $\left\{\begin{array}{l}4 \ln (x)+3 y^{2}=1 \\ 3 \ln (x)+2 y^{2}=-1\end{array}\right.$
|
$\left(e^{-5}, \pm \sqrt{7}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(e^{-5}, \pm \sqrt{7}\right)
|
college_math.PRECALCULUS
|
exercise.10.7.37
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan ^{2}(x)=\frac{3}{2} \sec (x)$
|
$x=\frac{\pi}{2}, \frac{3 \pi}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{2}, \frac{3 \pi}{2}
|
college_math.PRECALCULUS
|
exercise.9.4.2
|
Simplify the expression: $\frac{10 !}{7 !}$
|
720
|
Creative Commons License
|
college_math.precalculus
|
720
|
college_math.PRECALCULUS
|
exercise.2.1.28
|
Jeff can walk comfortably at 3 miles per hour. Find a linear function $d$ that represents the total distance Jeff can walk in $t$ hours, assuming he doesn't take any breaks.
|
$d(t)=3 t, t \geq 0$.
|
Creative Commons License
|
college_math.precalculus
|
d(t)=3 t, t \geq 0$.
|
college_math.PRECALCULUS
|
exercise.5.3.17
|
Solve the equation or inequality: $x+1=\sqrt{3 x+7}$
|
$x=3$
|
Creative Commons License
|
college_math.precalculus
|
x=3
|
college_math.PRECALCULUS
|
exercise.6.4.16
|
Solve the equation analytically: $\log _{5}(2 x+1)+\log _{5}(x+2)=1$
|
$x=\frac{1}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{1}{2}
|
college_math.PRECALCULUS
|
exercise.6.3.45
|
Use your calculator to help you solve the inequality: $e^{x}<x^{3}-x$
|
$\approx(2.3217,4.3717)$
|
Creative Commons License
|
college_math.precalculus
|
\approx(2.3217,4.3717)
|
college_math.PRECALCULUS
|
exercise.10.7.104
|
Express the domain of the function using the extended interval notation: $f(x)=\frac{\sin (x)}{2+\cos (x)}$
|
$(-\infty, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, \infty)
|
college_math.PRECALCULUS
|
exercise.2.2.14
|
Solve the equation: $|x|=12-x^{2}$
|
$x=-3$ or $x=3$
|
Creative Commons License
|
college_math.precalculus
|
x=-3$ or $x=3
|
college_math.PRECALCULUS
|
exercise.5.3.40
|
The period of a pendulum in seconds is given by
$$
T=2 \pi \sqrt{\frac{L}{g}}
$$
(for small displacements) where $L$ is the length of the pendulum in meters and $g=9.8$ meters per second per second is the acceleration due to gravity. My Seth-Thomas antique schoolhouse clock needs $T=\frac{1}{2}$ second and I can adjust the length of the pendulum via a small dial on the bottom of the bob. At what length should I set the pendulum?
|
$9.8\left(\frac{1}{4 \pi}\right)^{2} \approx 0.062$ meters or 6.2 centimeters
|
Creative Commons License
|
college_math.precalculus
|
9.8\left(\frac{1}{4 \pi}\right)^{2} \approx 0.062$ meters or 6.2 centimeters
|
college_math.PRECALCULUS
|
exercise.6.5.12
|
A finance company offers a promotion on $\$ 5000$ loans. The borrower does not have to make any payments for the first three years, however interest will continue to be charged to the loan at $29.9 \%$ compounded continuously. What amount will be due at the end of the three-year period, assuming no payments are made? If the promotion is extended an additional three years, and no payments are made, what amount would be due?
|
$A(3)=5000 e^{0.299 \cdot 3} \approx \$ 12,226.18, A(6)=5000 e^{0.299 \cdot 6} \approx \$ 30,067.29$
|
Creative Commons License
|
college_math.precalculus
|
A(3)=5000 e^{0.299 \cdot 3} \approx \$ 12,226.18, A(6)=5000 e^{0.299 \cdot 6} \approx \$ 30,067.29
|
college_math.PRECALCULUS
|
exercise.7.2.13
|
Find the standard equation of the circle which satisfies the given criteria: center $(3,5)$, passes through $(-1,-2)$
|
$(x-3)^{2}+(y-5)^{2}=65$
|
Creative Commons License
|
college_math.precalculus
|
(x-3)^{2}+(y-5)^{2}=65
|
college_math.PRECALCULUS
|
exercise.9.1.9
|
Write out the first four terms of the given sequence: $b_{1}=2, b_{k+1}=3 b_{k}+1, k \geq 1$
|
$2,7,22,67$
|
Creative Commons License
|
college_math.precalculus
|
2,7,22,67
|
college_math.PRECALCULUS
|
exercise.6.3.41
|
Use your calculator to help you solve the equation: $e^{x}=\ln (x)+5$
|
$x \approx 0.01866, x \approx 1.7115$
|
Creative Commons License
|
college_math.precalculus
|
x \approx 0.01866, x \approx 1.7115
|
college_math.PRECALCULUS
|
exercise.3.3.43
|
Find the real solutions of the polynomial equation $14 x^{2}+5=3 x^{4}$.
|
$\{-2\} \cup[1,3]$
|
Creative Commons License
|
college_math.precalculus
|
\{-2\} \cup[1,3]
|
college_math.PRECALCULUS
|
exercise.6.4.30
|
Solve the inequality analytically: $\ln \left(x^{2}\right) \leq(\ln (x))^{2}$
|
$(0,1] \cup\left[e^{2}, \infty\right)$
|
Creative Commons License
|
college_math.precalculus
|
(0,1] \cup\left[e^{2}, \infty\right)
|
college_math.PRECALCULUS
|
exercise.6.4.15
|
Solve the equation analytically: $\log _{3}(x-4)+\log _{3}(x+4)=2$
|
$x=5$
|
Creative Commons License
|
college_math.precalculus
|
x=5
|
college_math.PRECALCULUS
|
exercise.10.7.10
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cos \left(x+\frac{5 \pi}{6}\right)=0$
|
$x=-\frac{\pi}{3}+\pi k ; x=\frac{2 \pi}{3}, \frac{5 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{\pi}{3}+\pi k ; x=\frac{2 \pi}{3}, \frac{5 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.1.3.40
|
Determine whether or not the equation represents $y$ as a function of $x$: $y=x^{2}+4$
|
Function
|
Creative Commons License
|
college_math.precalculus
|
Function
|
college_math.PRECALCULUS
|
exercise.10.7.39
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan (2 x)-2 \cos (x)=0$
|
$x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \frac{3 \pi}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \frac{3 \pi}{2}
|
college_math.PRECALCULUS
|
exercise.10.7.107
|
Express the domain of the function using the extended interval notation: $f(x)=\arcsin (\tan (x))$
|
$\bigcup_{k=-\infty}^{\infty}\left[\frac{(4 k-1) \pi}{4}, \frac{(4 k+1) \pi}{4}\right]$
|
Creative Commons License
|
college_math.precalculus
|
\bigcup_{k=-\infty}^{\infty}\left[\frac{(4 k-1) \pi}{4}, \frac{(4 k+1) \pi}{4}\right]
|
college_math.PRECALCULUS
|
exercise.10.7.63
|
Solve the equation: $4 \operatorname{arcsec}\left(\frac{x}{2}\right)=\pi$
|
$x=2 \sqrt{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=2 \sqrt{2}
|
college_math.PRECALCULUS
|
exercise.6.3.5
|
Solve the equation analytically: $8^{x}=\frac{1}{128}$
|
$x=-\frac{7}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{7}{3}
|
college_math.PRECALCULUS
|
exercise.1.1.23
|
Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(3,-10),(-1,2)$
|
$d=4 \sqrt{10}, M=(1,-4)$
|
Creative Commons License
|
college_math.precalculus
|
d=4 \sqrt{10}, M=(1,-4)
|
college_math.PRECALCULUS
|
exercise.6.3.13
|
Solve the equation analytically: $(1.005)^{12 x}=3$
|
$x=\frac{\ln (3)}{12 \ln (1.005)}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\ln (3)}{12 \ln (1.005)}
|
college_math.PRECALCULUS
|
exercise.10.7.19
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (x)=\cos (x)$
|
$x=\frac{\pi}{4}, \frac{5 \pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{4}, \frac{5 \pi}{4}
|
college_math.PRECALCULUS
|
exercise.1.3.35
|
Determine whether or not the equation represents $y$ as a function of $x$: $x^{3} y=-4$
|
Function
|
Creative Commons License
|
college_math.precalculus
|
Function
|
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