Maximum value of (log x)/x

To solve for x, given that the perimeter of a square is equal to the perimeter of an equilateral triangle, the equation 4x = 3(x + 1) can be used.

To find the value of x when the perimeter of a square is equal to the perimeter of an equilateral triangle, we set up an equation. Since the side of the square is given by x, and the perimeter of a square is 4 times the length of a side, the perimeter of the square is 4x. The length of a side of the equilateral triangle is given by x + 1, and since the perimeter of the equilateral triangle is 3 times the length of a side, its perimeter is 3(x + 1). Setting these two perimeters equal to each other gives us the equation 4x = 3(x + 1).

Keywords

parallel, perpendicular, graphing, linear equation, slope, lines

we know that

If two lines are perpendicular, then the product of their slope is equal to minus one

so

[tex]m1*m2=-1[/tex]

If two lines are parallel, then their slope are equal

[tex]m1=m2[/tex]

In this problem we have

[tex]y=x+11[/tex] --------> linear equation A

the slope is [tex]m1=1[/tex]

[tex]y=-x+2[/tex]  --------> linear equation B

the slope is [tex]m2=-1[/tex]

Find the product

[tex]m1*m2=-1[/tex] ---------> the lines are perpendicular

therefore

the answer is the option B

Perpendicular

using a graphing tool

see the attached figure

Answer:

Perpendicular

Step-by-step explanation:

Given are two equations as

[tex]y=x+11\\y=-x+2[/tex]

We have to find whether these two are parallel or perpendicular or neither

For this first we have to find the slope of these two lines

I line slope = 1

II line slope =-1

Since slopes are not equal, the lines are not parallel.

Let us check product of these slopes.  IF product =-1 the lines are perpendicular

We find that product = [tex]1*-1=-1[/tex]

Hence two lines are perpendicular

Answer: X= 55

Step-by-step explanation:

The sum of exterior angles of any polygon will always equal to 360 degrees. which means, we can write:

360 = (x+10)  +  (2x+20) + 3x  

x = 55

The exterior angles have values of 65, 130 and 165. The exterior that will have the largest value will correspond to the smallest interior angle and therefore  have a value of 15 degree!.  ;)

Final answer:

By calculating the value of x and using the relationship that each exterior angle and its corresponding interior angle are supplementary, we find that the measure of the smallest interior angle of the triangle is 115 degrees.

Explanation:

To find the measure of the smallest interior angle of the triangle, we first need to use the fact that the sum of the exterior angles of a triangle is always 360 degrees. So, we set up the equation (x+10) + (2x+20) + 3x = 360 and solve for x. This gives us:

x + 10

2x + 20

3x

Combining like terms, we get 6x + 30 = 360. Subtracting 30 from both sides, we get 6x = 330. Dividing both sides by 6, we find that x = 55 degrees. Therefore, the exterior angles are 65 degrees, 130 degrees, and 165 degrees. To find the smallest interior angle, we take the smallest exterior angle, 65 degrees and subtract it from 180 degrees (since the exterior angle and interior angle at the same vertex are supplementary). Thus, the smallest interior angle is 180 degrees - 65 degrees = 115 degrees.

By setting the upstream and downstream distances equal and solving for the time rowed upstream, we find that the fisherman rowed 4.2 hours upstream and 2.8 hours downstream. Since he rowed back to his starting point, the total distance rowed upstream and downstream is the same, amounting to 33.6 miles for the entire trip.

To solve this problem, we need to determine how much time the fisherman spent rowing upstream and downstream. Let's call the upstream speed U, which is 4 mph, and the downstream speed D, which is 6 mph. He rows for a total of 7 hours. Let's call the time spent rowing upstream T and the time spent rowing downstream (7 - T) since the entire trip takes 7 hours.

Since distance equals rate times time, the upstream distance will be U × T and the downstream distance will be D × (7 - T). Because the fisherman returns to his starting point, the upstream and downstream distances are equal.

Setting them equal gives us the equation:
4T = 6 (7 - T)

Solving for T gives us:
T = 42 / 10 = 4.2 hours

Hence, the time spent rowing downstream is 7 - 4.2 = 2.8 hours.

To find the total distance rowed, we use the downstream distance (since it's equal to the upstream distance) which is calculated as:
Distance = D × (7 - T) = 6 × 2.8 = 16.8 miles

Therefore, the fisherman rowed a total of 16.8 miles upstream and 16.8 miles downstream, summing up to 33.6 miles for the entire trip.

The probability of a fisherman catching a trout first and then a second trout from this lake is 0.158.

The subject of this question pertains to the field of probability in mathematics and requires an understanding of fundamental concepts related to this topic.

      0.4 x (79/199) = 0.158.

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To find the maximum profit, take the derivative of the profit function and set it equal to zero, then solve for s. This gives the advertising spend that yields the maximum profit.

To find the amount of money the company should spend on advertising in order to yield a maximum profit, we first need to find the derivative of the profit function P(s) = -(1/10)s^3 + 6s^2 + 400. The derivative, P'(s), gives the rate of change of profit with respect to advertising spending.

The derivative of the function is P'(s) = -(3/10)s^2 + 12s. We find the maximum of the function by setting the derivative equal to zero and solving for s.

Setting P'(s)=0 we get: -(3/10)s^2 + 12s = 0

Solving this equation will provide the advertising spend for maximum profit.

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Final answer:

To find the coefficient of 1/x in the expansion of (2x - 1/x)^5, we use the binomial theorem, specifically the term where the exponent of x is -1. We find that the coefficient of 1/x is -80.

Explanation:

To find the coefficient of 1/x in the expansion of (2x - 1/x)^5, we can use the binomial theorem. The binomial theorem states that:

(a + b)^n = Σ (n choose k) a^(n-k) b^k

where 'n choose k' is the binomial coefficient, computed as:

n! / [(n-k)!k!]

In our case, a = 2x and b = -1/x, and we want the term where the power of x is -1, which occurs when k = 4:

(5 choose 4) (2x)^(5-4) (-1/x)^4 = 5 * 2 * 1 * 1/x^4 = 10/x^4

However, to get the term with 1/x, we need a positive power of 1 for x. We achieve this when k = 1:

(5 choose 1) (2x)^(5-1) (-1/x)^1 = 5 * 16x^4 * (-1/x) = -80

Therefore, the coefficient of 1/x in the expansion is -80.

Answer:

We must have two angles and a side.

A is the correct option.

Step-by-step explanation:

For any triangle ABC,  the law of sine is given by

[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]

From this formula it is clear that in order to find the length of the side of the triangle, we must have two angles and a side.

Let us understand this by assuming that we need to find a (length of the side). From the formula, we have

[tex]\frac{a}{\sin A}=\frac{b}{\sin B}\\\\a=\frac{b\sin A}{\sin B}[/tex]

Thus, to find the length a, we must have b, sin A and sin B.

Hence, o find the length of the side of the triangle, we must have two angles and a side.

Given that two numbers sum to 's', and one of the numbers is 'n', the other number can be expressed as 's - n'. In essence, if you subtract 'n' from the sum 's', you obtain the second number.

The student's question relates to expressing one number in terms of another when their sum is given. Given that the sum of the two numbers is s and one of the numbers is n, the second number can be expressed as s - n. This is because when two numbers add to give 's', subtracting one of the numbers (n, in this case) from 's' would give the other number.

For instance, if s = 10 and n = 3, then the second number would be 10 - 3 = 7. Hence, the correct answer from the options provided would be 3. s - n.

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To find the standard form equation of a hyperbola with given vertices and foci, calculate the values of a and b using distances from the origin, and then insert into the standard form equation (y²/a²) - (x²/b²) = 1. The final equation for the hyperbola with vertices at (0, ±2) and foci at (0, ±7) is 45y² - 4x² = 180.

To find the equation of a hyperbola in standard form, we need the vertices and the foci. Given that the vertices are at (0, ±2), we know that the hyperbola is vertical, and the distance between the vertex and the center, which is the origin, is 2 units, so a=2. Since the foci are at (0, ±7), the distance from the center to a focus is 7 units, which means c=7.

With these values, we can calculate b by using the relationship c² = a² + b². Plugging in the known values gives us 7² = 2² + b², so b² = 49 - 4 = 45. Therefore, b = √45.

The standard form equation of a vertical hyperbola with a center at the origin is (y²/a²) - (x²/b²) = 1. Replacing a and b with the found values, the equation becomes (y²/4) - (x²/45) = 1. Multiplying through by 180, which is the least common multiple of 4 and 45, we get the equation in standard form: 45y² - 4x² = 180.

Final answer:

The area of a square as a function of its perimeter is A = (P/4)².

Explanation:

The area of a square can be represented as a function of its perimeter. Let's say the side length of the square is a. The perimeter of the square is given by P = 4a. So, to write the area A as a function of the perimeter P, we can use the formula:
A = (P/4)²

Answer: 4,2 is the correct answer

Step-by-step explanation:

Answer:

1: 720

2: [tex]\frac{1}{336}[/tex]

Step-by-step explanation:

1: There are 10 kids in a race so the number of ways such that they come first , second and third=10×9×8

                           =720

(since the first place is occupied by any one of the ten students i.e. we have ten choices after occupying first position there are only 9 students left for second position and after occupying second for the third position only 8 choices are left)

2: We have to find the probability such that there is no replacement  such that we draw 135.

for the first position we have to find the probability([tex]P_{1}[/tex]) that '1' comes out of 8 numbers=[tex]\frac{1}{8}[/tex]

similarly for second choice we could not take 1 as there is no replacement

so probability that 3 comes([tex]P_{2}[/tex]) =[tex]\frac{1}{7}[/tex]

similarly probability that 5 comes at third position([tex]P_{3}[/tex]) = [tex]\frac{1}{6}[/tex]

hence, probability that 135 is drawn= [tex]\frac{1}{8}\times\frac{1}{7}\times\frac{1}{6}=\frac{1}{336}[/tex]

Solving literal equations for a variable is beneficial for finding general solutions that apply to multiple situations, saves time by avoiding repetitive calculations, and reduces potential errors when working with complex numbers.

Why Solve Literal Equations for a Variable

Solving a literal equation for a variable is fundamental in mathematics because it allows one to find a general solution that can apply to various situations. Instead of replacing a variable with a given number each time, which can be time-consuming and error-prone especially when dealing with long decimals and complex fractions, working with no numbers—just variables—simplifies the process. Having found the values that make the equation true, one can easily convert words to an algebraic expression or equation that best models a given problem. This results in the ability to find a solution more swiftly and with less room for error.

For example, in computer programs, solutions to equations are frequently required for multiple sets of input values. Algebraically solving an equation provides a general form that can easily be reused with different inputs, saving substantial time and resources. Additionally, in complex problems involving multiple unknowns, solving for one variable in terms of others can lead to a simplified equation, making it easier to solve the problem.

Answer:

a horizontal translation and a glide reflection

Step-by-step explanation:

Answer:

[tex]y-2=3(x-4)[/tex]

Step-by-step explanation:

We are given that a line has a slope (m) of 3 and passes through (4, 2). We want to write this equation in point-slope form.

Point-slope form is given as [tex]y-y_1=m(x-x_1)[/tex] where m is the slope and [tex](x_1,y_1)[/tex] is a point.

We can substitute what we know into the equation.

First, substitute 3 as m.

[tex]y-y_1=3(x-x_1)[/tex]

Now, substitute 4 as [tex]x_1[/tex] and 2 as [tex]y_1[/tex].

[tex]y-2=3(x-4)[/tex]

Answer:

(8.33) , (236.67)

Step-by-step explanation:

This is correct