Date: Pages of * The current ratio of a business is the ratio of its current assets to its current labilities. After consulting with the comptroller, the president of Ace sports equipment company decides to take out a short-term loan to build up inventory. The company has current assets of $350,000 and current labilities of $ 80,000. How much can the company borrow if the current ratio is to be no less than 2.5?
​

Answer:

1.a)  x = g(2)

1.b)  See below

2.a)  See below

2.b) α = 60°

3.a)  2 + h

3.b)  2

4.a)  Interval [3, 4] = 3 m/s

       Interval [3, 3.1] = 2.1 m/s

4.b)  2 m/s

Step-by-step explanation:

Part (a)

Given function f(x):

\(f(x)=\dfrac{x}{x-2}\)

If the range of f is (-∞, 1) U (1, ∞), then a value of y in the range of f is y=2.

Set the function equal to the value of y and solve for x:

\(\begin{aligned}y=2 \implies \dfrac{x}{x-2}&=2\\x&=2(x-2)\\x&=2x-4\\-x&=-4\\x&=4\end{aligned}\)

The solution written as a function x = g(y) is:

Part (b)

Given function g(y):

\(g(y)=\dfrac{2y}{y-1}\)

Verify that g is the inverse function of f by calculating g(f(x)):

\(\begin{aligned}\implies g(f(x)) &=\dfrac{2f(x)}{f(x)-1}\\\\&=\dfrac{2\left(\frac{x}{x-2}\right)}{\left(\frac{x}{x-2}\right)-1}\\\\&=\dfrac{2\left(\frac{x}{x-2}\right)}{\left(\frac{x-(x-2)}{x-2}\right)}\\\\&=\dfrac{\left(\dfrac{2x}{x-2}\right)}{\left(\dfrac{2}{x-2}\right)}\\\\&=\dfrac{2x}{2}\\\\&=x \quad (\text{for all $x\neq 2$})\end{aligned}\)

Similarly, calculate f(g(y)):

\(\begin{aligned}\implies f(g(y))&=\dfrac{g(y)}{g(y)-2}\\\\&=\dfrac{\left(\frac{2y}{y-1}\right)}{\left(\frac{2y}{y-1}\right)-2}\\\\&=\dfrac{\left(\frac{2y}{y-1}\right)}{\left(\frac{2y-2(y-1)}{y-1}\right)}\\\\&=\dfrac{\left(\dfrac{2y}{y-1}\right)}{\left(\dfrac{2}{y-1}\right)}\\\\&=\dfrac{2y}{2}\\\\&=y \quad \text{(for all $y \neq 1$)}\end{aligned}\)

Hence verifying that g is the inverse function of f.

\(\boxed{\begin{minipage}{5 cm}\underline{Trigonometric Identities}\\\\$\tan x=\dfrac{\sin x}{\cos x}$\\\\\\$\cos \left(\arctan \left(x\right)\right)=\dfrac{1}{\sqrt{1+x^2}}$\\\\\\$\sin\left(\arctan \left(x\right)\right)=\dfrac{x}{\sqrt{1+x^2}}$\\\end{minipage}}\)

Part (a)

If  y = arctan(x)  then  tan(y) = x.

Use the trigonometric identities to express sin(y) and cos(y) in terms of x.

\(\boxed{\begin{aligned}\tan(y) &= x\\\implies \dfrac{\sin y}{\cos y}&=x\\\sin y&=x \cos y\\ \sin y&=x \cos (\arctan x)\\ \sin y&=\dfrac{x}{\sqrt{1+x^2}}\end{aligned}}\)

\(\boxed{\begin{aligned}\tan(y) &= x\\\\\implies \dfrac{\sin y}{\cos y}&=x\\\\\dfrac{\sin y}{x}&= \cos y\\\\ \cos y&=\dfrac{\sin(\arctan x)}{x}\\\\ \cos y&=\dfrac{\dfrac{x}{\sqrt{1+x^2}}}{x}\\\\\cos y&=\dfrac{x}{\sqrt{x^2+1}x}\\\\\cos y&=\dfrac{1}{\sqrt{x^2+1}}\end{aligned}}\)

Part (b)

Given linear equation:

\(y=\sqrt{3}x+1\)

As the slope of a linear equation y = mx + b is m, then the slope (m) of the given line is √3.

If m = tan(α) then:

\(\begin{aligned}\implies m&=\sqrt{3}\\\tan (\alpha)&=\sqrt{3}\\\alpha&=\arctan \sqrt{3}\\\alpha&=60^{\circ}+180^{\circ}n\\\end{aligned}\)

If α is the angle the line forms with the x-axis, then α = 60°.

Given function:

\(f(x)=x^2-1\)

Part (a)

\(\begin{aligned}\dfrac{f(1+h)-f(1)}{h} &=\dfrac{((1+h)^2-1)-(1^2-1)}{h}\\\\&=\dfrac{(1+2h+h^2-1)-0}{h}\\\\&=\dfrac{2h+h^2}{h}\\\\&=2+h\end{aligned}\)

Part (b)

Part a used differentiating from first principles to find the gradient of f(x) at x = 1.  As h gets smaller, the gradient of the straight line gets closer and closer to the gradient of the curve.  Therefore, as h gets close to zero, (2 + h) gets close to 2.  Therefore, the slope of the tangent line to the graph of f(x) at the point where x = 1 is 2.

A particle is moving on the x-axis and its position at time is given by

\(x(t)=t^2-4t+3\)

Evaluate the position of the particle at t = 3, t = 4 and t = 3.1:

\(x(3)=3^2-4(3)+3=0\)

\(x(4)=4^2-4(4)+3=3\)

\(x(3.1)=(3.1)^2-4(3.1)+3=0.21\)

Average velocity formula

\(\overline{v}=\dfrac{s(t_2)-s(t_1)}{t_2-t_1}\)

Therefore, the average velocity of this particle over the interval [3, 4]:

\(\overline{v}=\dfrac{x(4)-x(3)}{4-3}=\dfrac{3-0}{1}=3\;\; \rm m/s\)

The average velocity over the interval [3, 3.1] is:

\(\overline{v}=\dfrac{x(3.1)-x(3)}{3.1-3}=\dfrac{0.21-0}{0.1}=2.1\;\; \rm m/s\)

Part (b)

To find an equation for velocity, differentiate the given equation for displacement:

\(v(t)=\dfrac{\text{d}x}{\text{d}t}=2t-4\)

To calculate the instantaneous velocity of the particle at time t = 3, substitute t = 3 into the equation for velocity:

\(v(3)=2(3)-4=2\;\; \rm m/s\)

Answer:

y-intercept = 2

Step-by-step explanation:

The y-intercept of a graph is simply the point at which the value of y, when x = 0. It is the point at which the y-axis is intercepted by the line.

In the graph provided, the line cuts the y-axis at y = 2, when x = 0.

Therefore the y-intercept of the graph provided = 2.

Answer:

the probability of rolling a sum of 6 with these dice is 1/6.

Step-by-step explanation:

To find the probability of rolling a sum of 6 with the given pair of dice, we can first list all possible pairs of outcomes that add up to 6:

(2,4)

(3,3)

(4,2)

For each of these pairs, we need to find the probability of rolling each number on its respective die and then multiply those probabilities together. The probability of rolling a particular number on one die is the number of times that number appears on that die divided by the total number of outcomes on that die.

For the first pair (2,4), the probability is:

(2 appears twice on one die out of six possible outcomes) × (4 appears once on the other die out of six possible outcomes) = (2/6) × (1/6) = 1/18

For the second pair (3,3), the probability is:

(3 appears twice on one die out of six possible outcomes) × (3 appears twice on the other die out of six possible outcomes) = (2/6) × (2/6) = 4/36

For the third pair (4,2), the probability is:

(4 appears twice on one die out of six possible outcomes) × (2 appears twice on the other die out of six possible outcomes) = (2/6) × (2/6) = 4/36

The total probability of rolling a sum of 6 is the sum of the probabilities of each possible pair:

1/18 + 4/36 + 4/36 = 1/6

Therefore, the probability of rolling a sum of 6 with these dice is 1/6.

The coordinates of point Q and point R are; (5, -6) and (-5, 6) respectively.

From the coordinates of point P as given to be;

When the point P is reflected across the y axis to obtain point Q; the polarity of its x-ordinate changes;

When the point P is reflected across the x-axis to obtain point R; the polarity of its y-ordinate changes;

Read more on reflection of points;

https://brainly.com/question/14062035

Comparing the √17 and 29/7 using <, > or = , we have √17 < 29/7.

Mathematical number comparison is the process or method of comparing two numbers to determine whether one is less, bigger, or equal to the other. The comparison symbols for numbers are "=", which stands for "equal to," "=", which stands for "greater than," and " ", which stands for "less than." Comparing amounts is a technique for figuring out how much to compare units in relation to a different standard or reference unit. A common reference point is necessary for two comparing units to be compared; otherwise, they cannot be compared.

Given,

Comparing the following using <, > or = ,

Numbers:

√17 and 29/7

For √17

Simplifying,

√17 = 4.123

For 29/7

29/7 = 4.14

√17 < 29/7

To learn more about comparing, visit:

https://brainly.com/question/29467965

#SPJ1

Answer:

26

Step-by-step explanation:

One leg measures 7.

One leg measures 8.

a² + b² = c²

7² + 8² = c²

c² = √113 = 10.630 = 11

perimeter = 7 + 8 + 11

perimeter = 26

The length of the circle is approximately 9.237 cm.

What are the diagonals of a rhombus?
A rhombus is a quadrilateral with all sides of equal length. It also has the following properties regarding its diagonals:

The diagonals bisect each other at 90 degrees.

The diagonals are perpendicular to each other.

The diagonals have the same length.

The diagonals divide the rhombus into four congruent triangles.

In a rhombus with a side of 8 cm and an angle of 60 degrees, the diagonals are equal in length.

The length of the diagonals can be found using the formula:

\($D_1 = \frac{2a}{\sqrt{3}}$\)

where a is the length of the side. Substituting the value of a as 8 cm, we get:

\($D_1 = \frac{2(8)}{\sqrt{3}} \approx 9.237 \text{ cm}$\)

The length of the circle inscribed in the rhombus is equal to the length of the diagonals.

Therefore, the length of the circle is approximately 9.237 cm.

To learn more about the diagonals of a rhombus visit:

https://brainly.com/question/30763485

#SPJ1

Answer:

im not sure of the answer but hopefully this will help

Step-by-step explanation:

basically the intervals in which its increasing and decreasing depend on the leading coefficient and the vertex

since this is opening up, the intervals will be

the function is increasing (x coord of vertex, infinity)

the function is decreasing (-infinity, x-coord of vertex)

Answer:

25 * 30= x

x= total number of candies

Answer: the answer is $175

Step-by-step explanation:

250*0.3 is 75

250-75 gives us 175

The required Comparison of the inequalities are

The range of values encompassing the region's junction is (-13, 15).

When comparing two numbers, an inequality indicates whether one is less than, larger than, or not equal to the other.

We take into account the various variables of the inequality

|x – 1| + 1 > 15

Therefore

|-x-1|+1-1<15-1

|-x-1|-1 <14

13<x<15

The required region lies between the inequality -13 <x< 15.

Simplify the inequality Ix-11+1 > 15 we get,

|x-1|+1 > 15

|x+1| +1-1 >15-1

|x-1| > 14

x> 15  

x<-13

Read more about inequalities

https://brainly.com/question/20383699

#SPJ1

Answer:

y = 12 x = 12\(\sqrt{3}\)

Step-by-step explanation:

This is a 60, 90, 30. It's a special triangle.

2z = 24

z = 12

If x = z\(\sqrt{3}\)

then x = 12\(\sqrt{3}\)

y = z itself

So y = 12

The y-coordinate value of Point C if the area of ABC is 9 units squared is: 5 or -1

Point A is at (x, 2) and B is at (x + 6, 2). Since AB must lie on the line y=2  and be 6 units long. Point C is on the line x = -3 . So let C be at (-3, y).

Since ΔABC is a right angle, then point C must have the same x-coordinate as point A. Therefore, A(-3, 2) and B(2, 2).

The area of ΔABC is 9. So,

9 = 1/2 (b)(h)

where b is the base and h is the height.

so b = 6 and h = AC

Solving this for AC gives

9 = 1/2 (6)(AC)

18/6 = AC

3 = AC

Point C must lie 3 units above point A or 3 units below the point A. If it lies 3 units above, then it has a y-coordinate of 2 + 3 = 5.

If it lies 3 units below, it has a y-coordinate 2 - 3 = -1.

Therefore, y-coordinate is 5 or -1.

Read more about segment coordinate at: https://brainly.com/question/12959377

#SPJ1

Answer:

Hello, The answer to your question is,

\(x=\) −\(\frac{2}{t^3}\)

hope this help :)
(correct me if i am wrong and im sorry)

Step-by-step explanation:

The calculated value of x in the equation (x + 1) (x - 5) = 16 is 7

From the question, we have the following parameters that can be used in our computation:

(x + 1) (x - 5) = 16

The above expression is the product of two factors

(x + 1) and (x - 5)

And the result is 16

Express 16 as 8 * 2

So, we have

(x + 1) (x - 5) = 8 * 2

By comparison, we have

x + 1 = 8 and x - 5 = 2

When evaluated, we have

x = 7 and x = 7

This means that the value of x in the equation is 7

Read more about equation at

https://brainly.com/question/148035

#SPJ1

Answer:

60×40%=24

Hope it helps!!!

Answer:

4/5 would be at the 8th line, and 3/10 would be on the 3rd

Step-by-step explanation:

Answer:

B)no

Step-by-step explanation:

the correct solution is (3,12)

Answer:

The odds are 100 out of 200% for both to have heads.

The actual probility is 50%

Step-by-step explanation:

It is nominal because they all mean the same thing

Answer:

k= 80%

Step-by-step explanation:

Jar A contains 4*0.45 L acid, and 4 L of a solution  of acid.

Jar B contains 5*0.48 L acid., and 5 L of a solution of acid.

Jar C contains 1*k/100 = k/100 acid, and 1 L of a solution.

50% = 0.5

For jar A.

(2/3)*k/100 L acid  is added to jar A.

Now jar A contains   4*0.45 L + (2/3)*k/100 L acid, and it has (4+2/3)L of a solution.

L solute/L solution = 0.5

[4*0.45 L + (2/3)*k/100 L]/(4+2/3)L = 0.5

[1.8 + (2k/300)]/[(12+2)/3] = 0.5

[1.8 + (2k/300)]/[14/3] = 0.5

[1.8 + (2k/300)]= 0.5*(14/3)

(2k/300) = 0.5*(14/3) - 1.8

2k = (0.5*(14/3) - 1.8)*300

k = (0.5*(14/3) - 1.8)*300/2 =80

k= 80%

We also can find k using jar B.

(1/3)k/100 L acid is added  to jar B.

Now jar B contains 5*0.48 L+ (1/3)k/100 L acid, and it has (5+1/3) L of a solution.

L solute/L solution = 0.5

[5*0.48 L+ (1/3)k/100 L ]/(5+1/3)L= 0.5

[5*0.48 + (1/3)k/100 ]/(5+1/3)= 0.5

This equation also gives k=80%

Check.

We can check at least for jar A.

Jar A has 4L solution and 4*0.45=1.8 L acid.

2/3 L of the solution from jar C was added, and now we have 4 2/3 L of solution.

(2/3)* 80%= (2/3)*0.8 acid was added from jar C.

Now we have [1.8 +(2/3)*0.8] L acid in jar A.

L solute/L solution =  [1.8 +(2/3)*0.8] L /(4 2/3) L = 0.5 or 50%  as it is given that jar A has 50% at the end.

subtraction [BODMAS 'S' comes last which means subtraction]

hope it helps...

Answer:

Subtraction

Step-by-step explanation:

The graphs of y = x + 1 and y = 2^x intersect at approximately (-0.3517, 0.6483) and (1.561, 3.561).

To find the points of intersection between the graphs of y = x + 1 and y = 2^x, we need to set the two equations equal to each other and solve for x.

Setting y = x + 1 equal to y = 2^x, we have:

x + 1 = 2^x

To solve this equation, we can use numerical methods or make observations to find the points of intersection. By observing the behavior of the two functions, we can see that they intersect at two points: one when x is negative, and another when x is positive.

For x < 0, the exponential function y = 2^x approaches 0 as x approaches negative infinity, while the linear function y = x + 1 continues to decrease as x becomes more negative. Thus, there is one point of intersection in this region.

For x > 0, the exponential function grows faster than the linear function, so there is another point of intersection in this region.

However, finding the exact values for the points of intersection requires numerical methods such as using a graphing calculator or solving the equation numerically. Approximate values for the points of intersection can be found as x ≈ -0.3517 and x ≈ 1.561.

for more such questions on graphs

https://brainly.com/question/29538026

#SPJ8

Answer:

t(2,-2),C(2,-5),z(5,-4),f(5,0)

Step-by-step explanation:

because it reflects across the x-axis the y changes to the opposite.

Step-by-step explanation:

because 0.25 = 1/4.

so, it is actually

82.5 + 8×(1/4)x = 338.5

which is then simplified

82.5 + (8/4)x = 338.5

82.5 + 2x = 338.5

Step-by-step explanation:

\(\displaystyle\\8(0.25x)=8(\frac{25}{10}x)=\frac{8*25}{100} x=\frac{200}{100}x=2x\)

Thus, 82.5+8(0.25x)=338.5  ≡  82.5+2x=338.5

Answer:

C: 4

Step-by-step explanation:

We can rearrange the numbers:

\(-16+20=+20-16=20-16=4\)

Answer:

C. 4

Step-by-step explanation:

-16+20 equals 4 becase:

-16 is under zero, and if we add 20 to that, it will take 16 of that 20 to fill up the negative part, and the rest part remaining will be 4. So that four will add up to be the positive number!

Thank You! Please Mark Me Brainliest!

A sorting algorithm is defined as an algorithm that sorts the elements of a list. The most commonly used sequences are numerical sequence and lexicographical sequence, ascending or descending.

A sorting algorithm is used to reorder an array or list of elements according to an element comparison operator. Comparison operators are used to define a new order of elements in that data structure. Consider example, The above list of characters is sorted in ascending order by their value. That is, characters with lower values are placed before characters with higher values. It is important in real life because it supports the development of the scientific concept that things can be owned and organized into distinct groups (e.g. creatures, cars, weather types, etc.). We will now discuss some real-world examples of using sorting algorithms.

1) Your phone's contact list is sorted. This means that your data is organized in this way so you can easily access the contacts you want from your phone. That is, "I ordered."

2) Paper sorting : Imagine a teacher sorting students' work alphabetically by student name. This type of operation is similar to the functionality of sorting algorithms such as bucket sort. Sorting is more efficient process.

3) Traffic lights : Traffic lights are a good example of how algorithms are used in the real world around us. Next time you get stuck in a car at a red light, think about the algorithm that traffic lights run." Most traffic lights don't automatically switch between green, yellow, and red. This algorithm is a well-established turn-by-turn algorithm that directs traffic correctly. It's in order (it may not seem that way when you're sitting at a red light).

To learn more about sorting algorithm, refer:

https://brainly.com/question/14698104

#SPJ4

Step-by-step explanation:

are you having problems with something if so message this guy

Answer:

c + f = 5

120c + 1140f = 2640

f = 5 - c

120c + 1140(5 - c) = 2640

120c + 5700 - 1140c = 2640

1020c = 3060

c = 3 coach tickets

Step-by-step explanation: