Geometry problem below

Assuming what is meant by "infinite solutions" are infinite number of solutions of a logarithmic equation.  

[tex]\frac{1}{2}\log x^2 - \log \sqrt{x} - \log \sqrt{x} = 0[/tex]

and

[tex]\frac{1}{2}x-2\ln e^x=-\frac{3}{2}x[/tex]

Logarithmic equations with infinite solutions have graphs that look like dying-out exponentials. They can have infinite solutions because there are infinitely many values of y that satisfy the equation.

Examples of Logarithmic Equations with Infinite Solutions:
1. Logarithms to the base 10 (common logarithms):

In the equations below, y is the exponent to which 10 must be raised to equal x, so y is the common logarithm (log) of x.
x = 10^y
x = 10^y+1

2. Logarithms to the base e (natural logarithms):

In the equations below, y is the power to which e must be raised to equal x, so y is the natural logarithm (ln) of x.
x = e^y
x = e^y+1

Both of these equations have graphs that look like dying-out exponentials. They have infinite solutions because there are infinitely many values of y that satisfy the equation. Whenever the base is positive and not equal to 1, logarithmic equations can have infinite solutions.

Answer:

x = 1.3472 to the nearest ten thousandth.

Step-by-step explanation:

14^x + 1 = 36

14^x = 35

Taking logarithms:-

x ln 14 = ln 35

x = ln 35 / ln 14

= 1.3472.

The solution, rounded to the nearest ten-thousandth, is x = 1.5404.

To solve the equation [tex]14^x + 1 = 36[/tex], we first isolate the exponential term by subtracting 1 from both sides of the equation, which gives us [tex]14^x = 35.[/tex]

The next step is to take the logarithm of both sides of the equation.

We could use any base for the logarithm, but it's common to use base 10 or the natural logarithm base (e). In this case, let's use the natural logarithm:

[tex]ln(14^x) = ln(35)[/tex]

We can then use the property of logarithms which allows us to bring the exponent down as a multiplier:

[tex]x * times ln(14) = ln(35)[/tex]

Now, you can solve for x by dividing both sides by ln(14):

[tex]x = ln(35) / ln(14)[/tex]

Using a calculator, we find the quotient and then round to the nearest ten-thousandth:

x = 1.5404

This value is the solution to the original equation.

[tex]A_{\triangle}=\dfrac{1}{2}bh\\\\\text{We have}\ b=7cm,\ h=4cm\\\\\text{Substitute:}\\\\A_{\triangle}=\dfrac{1}{2}(7)(4)=\dfrac{28}{2}=\boxed{14\ cm^2}[/tex]

Answer:

y=0.5x-2

Step-by-step explanation:

if it is perpendicular to the line y=-2x-6, then you know that its slope is the negative reciprocal of that line, and it has a different y intercept which you need to solve for using the point given. You solve by plugging in the x and y values from the point and plugging in the slope into the standard equation, and solving for b, the y intercept

y=0.5x+b

1=0.5(6)+b

1=3+b

-2=b

The equation of the line that passes through the point (6, 1) and is perpendicular to the line whose equation is y=−2x−6  is y = 0.5x - 2.

From the classic definition of straight lines, we know that if we have to find an equation of a straight line being perpendicular to another straight line then the slope of the new equation of straight lines becomes negative reciprocal of the slope of given perpendicular line.

Mathematically, let m1 be the slope of the new straight line and m be the slope of the given perpendicular line, then we have

m1 = -(1/m)

Now, we have given equation y = -2x - 6

Thus slope of the required equation is say (m1) = -(-1/2) = 0.5

Thus the equation formed is y = (m1)x + c [where c is the y-intercept]

∴ y = 0.5x + c

The point given is (6,1) , thus y = 1 and x = 6

Thus the given equation can be formed as

⇒ 1 =  0.5*6 + c

∴  c = 1 - 0.5*6 = -2

The value of y-intercept of the required straight line is -2

The equation of straight line formed is y = 0.5x - 2.

Thus the equation of the line that passes through the point (6, 1) and is perpendicular to the line whose equation is y = − 2x − 6  is y = 0.5x - 2.

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Answer: 60 cans in each box

Step-by-step explanation:

98+82=180

180/3=60

[tex]\text{The domain}\\\\x\neq0\ \wedge\ x\neq-2\ \wedge\ x\neq2[/tex]

[tex]\dfrac{5}{x^2-4}+\dfrac{2}{x}=\dfrac{2}{x-2}\qquad\text{subtract}\ \dfrac{2}{x-2}\ \text{from obth sides}\\\\\dfrac{5}{x^2-2^2}+\dfrac{2}{x}-\dfrac{2}{x-2}=0\\\\\dfrac{5}{(x-2)(x+2)}+\dfrac{2}{x}-\dfrac{2}{x-2}=0\\\\\dfrac{5x}{x(x-2)(x+2)}+\dfrac{2(x-2)(x+2)}{x(x-2)(x+2)}-\dfrac{2x(x+2)}{x(x-2)(x+2)}=0\\\\\dfrac{5x+2(x^2-4)-2x(x+2)}{x(x-2)(x+2)}=0\\\\\dfrac{5x+2x^2-8-2x^2-4x}{x(x-2)(x+2)}=0\\\\\dfrac{x-8}{x(x-2)(x+2)}=0\iff x-8=0\to\boxed{x=8}\in D[/tex]

Answer:

So when you flip the coin 8 times and you get tails 3 of the times then you should do 3 plus 5 and you will get 8 as your total again so then the answer is 5 out of 8 total.

Step-by-step explanation:

The probability that all the remaining flips will be tails is 1/32, or approximately 0.03125.

The probability of getting tails on each coin flip is 50 percent since a fair coin has two equally likely outcomes: heads or tails.

If the first three coin flips resulted in tails, the remaining five coin flips are independent events. The probability of getting tails on each of the remaining flips is still 50 percent.

The probability of getting all the remaining flips to be tails is calculated by multiplying the probabilities of each individual flip. Since there are five remaining flips, the probability is 0.5 raised to the power of 5, or 1/32 (approximately 0.03125).

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Two supplementary angle when added together need to equal 180 degrees.

Answer:

Two supplementary angle when added together need to equal 180 degrees.

Step-by-step explanation:

Answer:  [tex]y -3 = \frac{4}{11} (x - 11)[/tex]

Step-by-step explanation:

We can use the point-slope formula to write the equation of a line given a point on the line and the slope of the line:  

Slope = m

Given point (x₁,y₁)

Formula = (y-y₁) = m(x-x₁)

Given point: (11,3); slope: 4/11

Answer : [tex]y - 3 = \frac{4}{11} (x - 11)[/tex]

[tex]\textit{\textbf{Spymore}}[/tex]​​​​​​

Answer:

the answer is a

Step-by-step explanation:

Answer:

B would be the answer for this question.

Step-by-step explanation:

Answer:  The fourth term is [tex]5000x^2.[/tex]

Step-by-step explanation:  We are given to find the fourth term in the expansion of the following binomial :

[tex]B=(2x+5)^5.[/tex]

We know that

the r-th term in the expansion of the binomial [tex](a+x)^n[/tex] is given by

[tex]T_r=^nC_ra^{n-(r-1)}b^{r-1}.[/tex]

For the given term, we have

n = 5  and  r = 4.

Therefore, fourth term is given by

[tex]T_4\\\\=^5C_{4-1}(2x)^{5-(4-1)}5^{4-1}\\\\=^5C_3(2x)^25^3\\\\=\dfrac{5!}{3!(5-3)!}\times4x^2\times125\\\\\\=\dfrac{5\times4}{2\times1}\times 500x^2\\\\=5000x^2.[/tex]

Thus, the fourth term is [tex]5000x^2.[/tex]

Answer:

The height of the object after 3 seconds is 45 feets.

Step-by-step explanation:

The height of an object thrown into the air with an initial upward velocity of 63 feet per second is given as

[tex]h(t)=-16t^2+63t[/tex]

Where, h(t) is height of the object in feet and t is the time in seconds.

We have to find the height of the object after 3 seconds. So, substitute t=3.

[tex]h(3)=-16(3)^2+63(3)[/tex]

[tex]h(3)=-16(9)+189[/tex]

[tex]h(3)=-144+189[/tex]

[tex]h(3)=45[/tex]

Therefore height of the object after 3 seconds is 45 feets.

To calculate the height of an object after 3 seconds using the equation h(t) = -16t^2 + 63t, substitute t with 3 and solve. This results in a height of 45 feet.

To find the height after 3 seconds, we substitute t with 3 into the equation, getting h(3) = -16(3)2 + 63(3). Calculating this step-by-step, we first find the square of 3, which is 9, and then multiply it by -16 to get -144. Next, we multiply 63 by 3 to get 189. Adding these two results together, we end up with 45 feet. Therefore, the height of the object after 3 seconds is 45 feet.

Answer:

[tex]I=12.5[/tex] amp

Step-by-step explanation:

The formula is,

[tex]E=\dfrac{P}{I}[/tex]

where,

E = Electromotive force in volts,

P = Power in watts,

I = Current in amps.

Given values are,

E = 3.6 volts,

P = 45 watts,

I = ??

Putting the values,

[tex]\Rightarrow 3.6=\dfrac{45}{I}[/tex]

[tex]\Rightarrow I=\dfrac{45}{3.6}=12.5[/tex] amp

Answer: Census.

Step-by-step explanation:

Given statement:- A research study is done to find the average age of all U.S. factory workers. The researchers asked every factory worker in Ohio what their birth year is.

This research is an example of a census because research in which information is obtained through the responses that all available members of an entire population give to questions.

In other words "Census is an official survey of population in a certain area and records various details about the individuals".

The study querying every factory worker in Ohio for their birth year to determine the average age of all U.S. factory workers is a census, as it attempts to gather data from every member of the entire population of interest. (First option)

The research study done to find the average age of all U.S. factory workers where the researchers asked every factory worker in Ohio their birth year is an example of a census. A census involves gathering information about every individual in the entire population of interest.

In this case, the population of interest would be all factory workers, and by querying every one of them (assuming it was indeed every single factory worker in Ohio), it constitutes a census, not a survey, which typically involves a representative sample.

It is not a convenience sample since that would imply a non-random selection based on ease of access, and it's not a simple random sample because not all members of the larger population (nationwide factory workers) have an equal chance of being included.

Answer:

A difference of squares has the following form [tex]a^2-b^2[/tex]. Any two perfect squares connected by subtraction can be factored.

It factors to (a+b)(a-b).

Step-by-step explanation:

A binomial is an expression with only terms where at least one is a term with a variable. When we can factor for difference of squares, we can have two variable terms or just one with a constant.

A difference of squares has the following form [tex]a^2-b^2[/tex]. Any two perfect squares connected by subtraction can be factored.

It factors to (a+b)(a-b).

Answer:

84°

Step-by-step explanation:

2 complementary angles sum to 90°

let x be the angle then complement = x - 78, hence

x + x - 78 = 90 ( add 78 to both sides )

2x = 168 ( divide both sides by 2 )

x = 84

hence the angle is 84°

Final answer:

Jupiter has 67 moons.

Explanation:

To solve this problem, let's define j as the number of moons Jupiter has. According to the given information, Jupiter has 11 more than 4 times as many moons as Neptune, which has 14 moons. So, we can set up an equation: j = 4n + 11, where n is the number of moons Neptune has. Since Neptune has 14 moons, we can substitute that value into the equation: j = 4(14) + 11. Simplifying further, we get j = 56 + 11 = 67. Therefore, Jupiter has 67 moons.

Answer:

C=25d

Step-by-step explanation:

We write an equation where C or cost is my output and d or days is my input. I should be able to put in any number of days and find the cost. Let's gather some data:

River Ramble

Day 1 $25(1)=25 cost

Day 2 $25(2)=50 cost

Day 3 $25(3)=75 cost

Day d $25(d)=C.

Our equation is C=25d.

Answer:

C

Step-by-step explanation:

The slope intercept of a line is y=mx +b where

This graph crosses the y-axis (the vertical line) halfway between -1 and -2. This  is -3/2. This means only answers a, c, and d are options.

The graph moves up from -3/2 to its next point at (-4,0). We calculate the slope using:

Slope:[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

We substitute [tex]x_1=0\\y_1=-1.5[/tex] and [tex]x_2=-4\\y_2=2[/tex]

[tex]m=\frac{2-(-1.5)}{-4-0}[/tex]

[tex]m=\frac{2+1.5}{-4}=\frac{3.5}{-4} =-0.875[/tex]

This decimal is equivalent to -7/8. This means C is the answer.

This equation represents a line with a slope of [tex]\( -\frac{7}{8} \)[/tex](meaning the line slopes downward from left to right) and a y-intercept of [tex]\( -\frac{3}{2} \)[/tex](where the line crosses the y-axis). The correct answer is option c

The slope-intercept form of the equation of a line is [tex]\( y = mx + b \),[/tex] where m represents the slope of the line, and b represents the y-intercept (where the line crosses the y-axis).

Let's analyze each option:

a.[tex]\( y = -\frac{8}{7}x - \frac{3}{2} \):[/tex]

  - Slope [tex]\( m = -\frac{8}{7} \)[/tex]

  - y-intercept [tex]\( b = -\frac{3}{2} \)[/tex]

b. [tex]\( y = -\frac{3}{2}x + \frac{7}{8} \):[/tex]

  - Slope [tex]\( m = -\frac{3}{2} \)[/tex]

  - y-intercept[tex]\( b = \frac{7}{8} \)[/tex]

c.[tex]\( y = -\frac{7}{8}x - \frac{3}{2} \):[/tex]

  - Slope [tex]\( m = -\frac{7}{8} \)[/tex]

  - y-intercept [tex]\( b = -\frac{3}{2} \)[/tex]

d.[tex]\( y = \frac{7}{8}x - \frac{3}{2} \):[/tex]

  - Slope[tex]\( m = \frac{7}{8} \)[/tex]

  - y-intercept[tex]\( b = -\frac{3}{2} \)[/tex]

Among the given options, the correct slope-intercept form is option c:

[tex]\[ \boxed{y = -\frac{7}{8}x - \frac{3}{2}} \][/tex]

Answer:

The simplified form of the given expression is 6-a.

Step-by-step explanation:

The given expression is

[tex](3-a)\cdot 2+a[/tex]

According to distributive property.

[tex]a\cdot (b+c)=ab+ac[/tex]

Use distributive property.

[tex]3(2)-a(2)+a[/tex]

[tex]6-2a+a[/tex]

Combine like terms.

[tex]6+(-2a+a)[/tex]

[tex]6-a[/tex]

Therefore the simplified form of the given expression is 6-a.

Answer:

It would equal 98.14

Step-by-step explanation:

Answer:

Function formula P(t) = -52t +480

Step-by-step explanation:

here, P(t) denotes the number of pages to read and t represents the time in hour.

Given the statement: Shryia read a 480-page-long book cover to cover in a single session, at a constant rate. After reading for 1.5 hours, she had 402 pages left to read.

⇒Total number of page in a long book = 480

After reading for 1.5 hours, she had 402 pages left to read.

Then,

Total number of page Shryia read in 1.5 hours is:

[tex]480-402 = 78[/tex]

Constant rate at which she is reading her book = [tex]\frac{78}{1.5} = 52[/tex] page per hour

Then, the function formula is given by:

P(t) = -52t + 480 ;  where t is in hours.

Check:

P(t) = -52t + 480

P(1.5) = -52(1.5) + 480 = -78 + 480 = 402     True.

Assuming this is for a programming language like c++, then the expression might look like

c = sqrt(a*a + b*b)

or you can use the pow function (short for power function)

c = sqrt( pow(a,2) + pow(b,2) )

writing "pow(a,2)" means "a^2"; similarly for b as well.

the length of hypotenuse can be found using the formula [tex]c = \sqrt{a^2+b^2}[/tex]

The pytharogas theorem states that:

[tex]hypotenuse^2 = perpendicular^2+ base^2[/tex]

The Pythagorean Theorem relates the length of the legs of a right triangle, labeled a and b, with the hypotenuse, labeled c. The relationship is given by:

[tex]a^2 + b^2 = c^2[/tex]

This can be rewritten, solving for c:

[tex]c = \sqrt{a^2+b^2}\\\\[/tex]

Thus, the length of hypotenuse can be found using the formula [tex]c = \sqrt{a^2+b^2}[/tex]

Answer:

Domain: [tex]( -\infty,\infty )[/tex] and Range: [tex][ -1,\infty )[/tex]

Step-by-step explanation:

We have the parametric equations [tex]x= 2t[/tex] and [tex]y=t^{2}+t+3[/tex].

Now, we will find the values of 'x' and 'y' for different values of 't'.

t                       :   -3       -2.5     -2     -1.5     -1     0       1      1.5      2

[tex]x= 2t[/tex]             :   -6       -5       -4       -3      -2    0      2      3        4

[tex]y=t^{2}+t+3[/tex]  :    9       6.75     5     3.75     3    3      5     6.75     9

Now, we can see that these parametric equations represents a parabola.

The general form of the parabola is [tex]y=ax^{2}+bx+c[/tex].

Now, we have the point ( x,y ) = ( 0,3 ). This gives that c = 3.

Also, we have the points ( x,y ) = ( -2,3 ) and ( 2,5 ). Substituting these in the general form gives us,

4a - 2b + 3 = 3 → 4a - 2b = 0 → b = 2a.

4a + 2b + 3 = 5 → 4a + 2b = 2 → 2a + b = 1 →  2a + 2a = 1 ( As, b = 2a ) → 4a = 1 → [tex]a=\frac{1}{4}[/tex].

So, [tex]b=\frac{1}{2}[/tex].

Therefore, the equation of the parabola obtained is [tex]y=\frac{x^{2}}{4}+\frac{x}{2}+3[/tex].

The graph of this function is given below and we can see from the graph that domain contains all real numbers and the range is [tex]y\geq -1[/tex].

Hence, in the interval form we get,

Domain is [tex]( -\infty,\infty )[/tex] and Range is [tex][ -1,\infty )[/tex]

Answer:

Domain:

[tex](-\infty,\infty)[/tex]

Range:

[tex][2.75,\infty)[/tex]

Step-by-step explanation:

we are given parametric equation as

[tex]x=2t[/tex]

[tex]y=t^2+t+3[/tex]

We can change into rectangular equation

we can eliminate t from first equation and plug into second equation

[tex]x=2t[/tex]

[tex]t=\frac{x}{2}[/tex]

now, we can plug that into second equation

[tex]y=(\frac{x}{2})^2+\frac{x}{2}+3[/tex]

now, we can draw graph

Domain:

we know that

domain is all possible values of x for which any function is defined

we can see that our equation is parabolic

and it is defined for all values of x

so, domain will be

[tex](-\infty,\infty)[/tex]

Range:

we know that

range is all possible values of y

we can see that

smallest y-value is 2.75

so, range will be

[tex][2.75,\infty)[/tex]

Answer:

852 miles

Step-by-step explanation:

We presume the speed is constant, so the satellite will travel 3/8 the distance in 3/8 the time.

... d = (3/8)·(2272 miles) = 852 miles

_____

1 minute is 1/8 of 8 minutes, so 3 minutes is 3/8 of 8 minutes.

Answer:

4 and 32

Step-by-step explanation:

We are given paired values for two variables x and y and we are to fill in the missing values such that they are in a proportional relationship.

For x, we have the following paired values:

[tex]1, 3, ___, 5, 8[/tex]

So from the given options, 4 fits the best here which is greater than 3 and lesser than 5.

And for y, we have:

[tex]4, 12, 16, 20, ___[/tex]

Here, 32 fits the best from all the options as it is the next (available) number after 20.