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3 years ago

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There are 31 students in a math class. Of those, 15 students have in-line skates, 10 have scooters, and 12 have skateboards. Ove

rall, 5 students have both in-line skates and a scooter, 3 have both a scooter and a skateboard, and 2 have both a skateboard and in-line skates. If no students have all three, how many students have only in-line skates, or only skateboards?

2 answers:

nekit [7.7K]3 years ago

7 0

Apecar has the correct answer, I checked it

masha68 [24]3 years ago

3 0

15 - In-line skates. 10 - Scooters. 12 - Skateboards.

5 have both skates and scooter: 15-5=10, 10-5=5.

10 - Skates. 5 - Scooters. 12 - Skateboards.

3 have both a scooter and skateboard: 5-3=2, 12-3=9.

10 - Skates. 2 - Scooters. 9 - Skateboards.

2 have skateboard and skates: 10-2=8, 9-2= 7.

Therefore, 8 students only have in-line skates and 7 have only skateboards. Hope this helped!

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What are the domain range of f(x)= 1/5 power of x

Snowcat [4.5K]

Answer:

Domain: all real numbers

Range: all real numbers

Step-by-step explanation:

I'm assuming you mean the function $f(x)=x^{\frac{1}{5}}$ . That's usually written

f(x) = x^(1/5) with the ^ meaning "to the power of..." and the fraction exponent in parentheses so as not to be confused with x^1/5 which could mean x to the first power, divided by 5.

Fractional exponents are used to indicate roots. In this case, x is being raised to the 1/5 power, so this is the fifth root of x, written $\sqrt[5]{x}$ . The 5 is called the root index.

For odd roots, like this one, the domain is all real numbers--<em>x</em> can be any number at all. So the domain is all real numbers.

The range is also all real numbers. Attached is a graph of this function. It might not look like it, but the graph rises to the right to any height. The larger <em>x</em> gets, the larger the 5th root gets. A similar thing happens on the left--the smaller <em>x</em> gets, the smaller the 5th root gets.

EDIT: see the comment. For the function $f(x)=(1/5)^x$ , the domain is all real numbers. The range is positive real numbers. I'll attach a graph!

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4 years ago

Find a second solution y2(x) of<br> x^2y"-3xy'+5y=0; y1=x^2cos(lnx)

rosijanka [135]

We can try reduction order and look for a solution $y_2(x)=y_1(x)v(x)$ . Then

$y_2=y_1v\implies{y_2}'=y_1v'+{y_1}'v\implies{y_2}''=y_1v''+2{y_1}'v+{y_1}''v$

Substituting these into the ODE gives

$x^2(y_1v''+2{y_1}'v+{y_1}''v)-3x(y_1v'+{y_1}'v)+5y_1v=0$

$x^2y_1v''+(2x^2{y_1}'-3xy_1)v'+(x^2{y_1}''-3x{y_1}'+5y_1)v=0$

$x^4\cos(\ln x)v''+x^3(\cos(\ln x)-2\sin(\ln x))v'=0$

which leaves us with an ODE linear in $w(x)=v'(x)$ :

$x^4\cos(\ln x)w'+x^3(\cos(\ln x)-2\sin(\ln x))w=0$

This ODE is separable; divide both sides by the coefficient of $w'(x)$ and separate the variables to get

$w'+\dfrac{\cos(\ln x)-2\sin(\ln x)}{x\cos(\ln x)}w=0$

$\dfrac{w'}w=\dfrac{2\sin(\ln x)-\cos(\ln x)}{x\cos(\ln x)}$

$\dfrac{\mathrm dw}w=\dfrac{2\sin(\ln x)-\cos(\ln x)}{x\cos(\ln x)}\,\mathrm dx$

Integrate both sides; on the right, substitute $u=\ln x$ so that $\mathrm du=\dfrac{\mathrm dx}x$ .

$\ln|w|=\displaystyle\int\frac{2\sin u-\cos u}{\cos u}\,\mathrm du=\int(2\tan u-1)\,\mathrm du$

Now solve for $w(u)$ ,

$\ln|w|=-2\ln(\cos u)-u+C$

$w=e^{-2\ln(\cos u)-u+C}$

$w=Ce^{-u}\sec^2u$

then for $w(x)$ ,

$w=Ce^{-\ln x}\sec^2(-\ln x)$

$w=C\dfrac{\sec^2(\ln x)}x$

Solve for $v(x)$ by integrating both sides.

$v=\displaystyle C_1\int\frac{\sec^2(\ln x)}x\,\mathrm dx$

Substitute $u=\ln x$ again and solve for $v(u)$ :

$v=\displaystyle C_1\int\sec^2u\,\mathrm du$

$v=C_1\tan u+C_2$

then for $v(x)$ ,

$v=C_1\tan(\ln x)+C_2$

So the second solution would be

$y_2=x^2\cos(\ln x)(C_1\tan(\ln x)+C_2)$

$y_2=C_1x^2\sin(\ln x)+C_2x^2\cos(\ln x)$

$y_1(x)$ already accounts for the second term of the solution above, so we end up with

$\boxed{y_2=x^2\sin(\ln x)}$

as the second independent solution.

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4 years ago

Rank the following values from largest to smallest.

Marrrta [24]

Assuming they are all standardised numbers, from largest:

Number of fans attending a World Series baseball game (10,000+)
Value of a home computer (~800)
Number of buttons on a telephone (10)
Average number of cars that can fit in a home garage (2)

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3 years ago

The graph of an equation with a negative discriminant always has what characteristic

Jet001 [13]

Answer:

See below.

Step-by-step explanation:

The zeroes of the equation will not be real so the graph will not pass through the x-axis.

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3 years ago

Write an equation in slope-intercept form of the line that passes through the given points.

Eddi Din [679]

Answer:

y = 3x + 2

Step-by-step explanation:

y = mx + b format

the y and x will stay the same

when graphed, the points make a line that intersects with the y-axis at 2. This would be the y - intercept or "b" in our equation.

m stands for the slope of the line which is represented by rise/run. Going left to right, the line is going upwards which means that the slope will be positive. the distance between these given points is rise 3 and run 1. 3/1

3/1 is the same thing as 3 so your slope would be 3.

your final equation would be y = 3x + 2

hope this helps :)

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3 years ago