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3 years ago

How to determine if a side length represents a right triangle?

Mathematics

1 answer:

jenyasd209 [6]3 years ago

5 0

Plug in each set of lengths into the Pythagorean theorem. Because the 2 sides are equal, that is a right triangle.

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Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible. Let x be the number and let

Allisa [31]

Answer:

$S(x) = x + \frac{1}{x}$ --- Objective function

Interval = $\{x:x=1\}$

Step-by-step explanation:

Given

Represent the number with x

The required sum can be represented as:

$x + \frac{1}{x}$

Hence, the objective function is:

$S(x) = x + \frac{1}{x}$

To get the the interval, we start by differentiating w.r.t x

<em>Using first principle, this gives:</em>

$S'(x) = 1 - \frac{1}{x^2}$

Equate S'(x) to 0 in order to solve for x

$0 = 1 - \frac{1}{x^2}$

Subtract 1 from both sides

$0 -1 = 1 -1 - \frac{1}{x^2}$

$-1 = - \frac{1}{x^2}$

Multiply both sides by -1

$1 = \frac{1}{x^2}$

Cross Multiply

$x^2 * 1 = 1$

$x^2 = 1$

Take positive square root of both sides because x is positive

$\sqrt{x^2} = \sqrt{1$

$x = 1$

Representing x using interval notation, we have

Interval = $\{x:x=1\}$

To get the smallest sum, we substitute 1 for x in $S(x) = x + \frac{1}{x}$

$S(1) = 1 + \frac{1}{1}$

$S(1) = 1 + 1$

$S(1) = 2$

<em>Hence, the smallest sum is 2</em>

3 0

2 years ago

Find the average value of the function over the given interval and all values of x in the interval for which the function equals

MrRissso [65]

The average value of $f(x)=4x^3-3x^2$ over [-1, 3] is

$\displaystyle\frac1{3-(-1)}\int_{-1}^3(4x^3-3x^2)\,\mathrm dx=\frac14\left(x^4-x^3\right)\bigg|_{-1}^3=\frac{(3^4-3^3)-(1+1)}4=\boxed{13}$

8 0

2 years ago

Which statements are true about exponential decay functions? Check all that apply. The domain is all real numbers. As the input

lorasvet [3.4K]

<h2>Answer:</h2>

The correct options are:

The domain is all real numbers.

The base must be less than 1 and greater than 0.

The function has a constant multiplicative rate of change.

<h2>Step-by-step explanation:</h2>

We know that the exponential function is given by:

$f(x)=ab^x$

where a>0 and b are constants.

Also, it represents a growth function if b>1

and a decay function if 0<b<1

where b is the base.

x belongs to whole of the real numbers( since the exponential function is well defined for all the real values of x.

Hence, the domain of the function is all the real numbers )

Also, the graph of a decay function decreases continuously i.e. with the increasing input value the output value decreases.

The exponential decay function always have a constant multiplicative rate of change i.e. b.

8 0

2 years ago

Read 2 more answers

In a fruit punch drink, the 3 ingredients are apple juice, orange juice and cranberry juice. If

Mila [183]

I would either way 3, or 4 idk but I don’t understand your question

5 0

2 years ago

Some students are working together to draw a picture of a playground.

vodomira [7]

From the information provided by the statement, you know that

*Ann draws 2/16 of the playground

*Susan draws 6/16 of the playground because she draws 3 times as much as Ann so

$3\cdot\frac{2}{16}=\frac{3}{1}\cdot\frac{2}{16}=\frac{6}{16}$

*Louise draws 3/16 because she draws half as much as Susan, so

$\frac{1}{2}\cdot\frac{6}{16}=\frac{6}{32}=\frac{2\cdot3}{2\cdot16}=\frac{3}{16}$

*Sam draws 2/16 because she draws 1 less section than Louise

$\frac{3}{16}-\frac{1}{16}=\frac{2}{16}$

Now, let x be the fraction of the playground that still needs to be drawn. Then, you have

$\begin{gathered} \frac{2}{16}+\frac{6}{16}+\frac{3}{16}+\frac{2}{16}+x=\frac{16}{16} \\ \text{ Add similar terms} \\ \frac{13}{16}+x=\frac{16}{16} \\ \text{ Subtract }\frac{13}{16}\text{from both sides of the equation} \\ \frac{13}{16}-\frac{13}{16}+x=\frac{16}{16}-\frac{13}{16} \\ x=\frac{3}{16} \end{gathered}$

Therefore, 3/16 of the playground still needs to be drawn.

5 0

5 months ago