It’s a hot summer day. Darius is babysitting and has promised his sweet little sister that he will fill her wading pool. He plans to soak his feet in the water as he sips a cool soda and watch little sister as she plays in the water. Sounds great, right? All he needs to do is get the pool filled. Unfortunately, little sister is impatient and starting to whine. Darius has a hose ready in the backyard to fill the pool, but he decides he will also drag the hose from the front-yard, so the two hoses can work together to fill the pool faster. Nobody wants their darling little sister to be unhappy!

Darius knows from experience that if he only uses one hose at a time, the front-yard hose takes 10 minutes longer to the fill the pool than the back-yard hose. With the two hoses working together, the pool is filled in 12 minutes. Once Darius is comfortable with his toes dangling in the water, he starts to wonder how long it would have taken to fill the pool if he had only used one of the hoses.

Darius thinks, “Hmmm. Usually when I think about how long it takes to do something I have to think about the rate it is going.”

5. Write a function, R(x), that models the combined rate of the two hoses in terms of the time that it takes for the backyard hose to fill the pool alone.

6.a. What is the entire domain of this function, if the context is not considered?

6.b. What is the domain of this function in the context of filling Darius’ pool? Explain your thinking after you submit.

7. Graph R(x). Label all important features and describe them.

Answer:

Step-by-step explanation:

home

The value of the function given as (f . g)(2) is D. (f.g)(2) = 44, the number of boats

It's important to note that functions are the expressions that are separated by an equal sign. It should be noted that that they've dependent as well as independent variables.

In this case, the function f(x)=2x²+3 represents the rate of a boat traveling with the current in miles per hour and the function g(x)=x+ 2 represents the time the boat traveled in hours. This will be illustrated as:

f(x) = 2x² + 3.

g(x) = x + 2

Therefore, we will evaluate the product

(f . g)(x) = (2x² + 3(x + 2)

(f . g)(x) = (2 × 2² + 3)(2 + 2)

(f . g)(2) = 44

The correct option is D.

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The function in f(t) form that describes the amount of fuel in the plane at time t will be;

⇒ f (t) = 112.25t + 31.425.75

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

After 13 minutes, the airplane has 32,882 gallons of fuel. After 31 minutes, the airplane has 34,902.5 gallons of fuel.

Now,

Since, After 13 minutes, the airplane has 32,882 gallons of fuel. After 31 minutes, the airplane has 34,902.5 gallons of fuel.

Hence, The equation of line passes through the points (13, 32882) and (31, 34902.5)

So, We need to find the slope of the line.

Hence, Slope of the line is,

m = (y₂ - y₁) / (x₂ - x₁)

m = (32882 - 34902.5) / ((31 - 13)

m = 2020.5 / 18

m = 112.25

Thus, The equation of line with slope 112.25 is,

⇒ y - 32885 = 112.25 (x - 13)

⇒ y - 32885 = 112.25x - 1459.25

⇒ y = 112.25x - 1459.25 + 32885

⇒ y = 112.25x + 31.425.75

Therefore, The function in f(t) form that describes the amount of fuel in the plane at time t will be;

⇒ f (t) = 112.25t + 31.425.75

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The length of shorter leg is 15 inches, length of longer leg is 36 inches and the length of hypotenuse is 39 inches.

What is right triangle?

A right triangle, also known as a right-angled triangle, an orthogonal triangle, or formerly a rectangled triangle, is a triangle with one right angle, meaning that its two sides are perpendicular. Trigonometry's foundation is the relationship between the right triangle's sides and other angle.

Let,

x = Longer leg of triangle

y = Shorter leg of triangle

h = Hypotenuse of triangle

Given: The length of the longer leg of a right triangle is 6inches more than twice the length of the shorter leg.

The length of the hypotenuse is 9inches more than twice the length of the shorter leg.

⇒  x = 6 + 2y

   h = 9 + 2y

The given triangle is a right angled triangle.

⇒                                 x^2 + y^2 = h^2

                         (6 + 2y)^2 + y^2 = (9 + 2y)^2

               36 + 24y + 4y^2 + y^2 = 81 + 36y + 4y^2

                         36 + 24y + 5y^2 = 81 + 36y + 4y^2

36 - 81 + 24y - 36y + 5y^2 - 4y^2 = 0

                            -45 - 12y + y^2 = 0

                              y^2 - 12y - 45 = 0

                      y^2 - 15y + 3y - 45 = 0

                       y(y - 15) + 3(y - 15) = 0

                                (y - 15)(y + 3) = 0

y - 15 = 0,   y + 3 = 0

y = 15,     y = -3

Since the length is not negative.

So, y = 15

Hence,

Shorter leg = 15 inchs

Longer leg = 6 + 2y = 6 + 2(15) = 36 inches

Hypotenuse = 9 + 2y = 9 + 2(15) = 39 inches

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Answer:

b

Step-by-step explanation:

Answer:

Step-by-step explanation:

for n=1 it is 7(1+2)=21

for n=2 it is 7(2+2)=28

for n=3 it is 7(3+2)=35

The range is ( 290, 230).

In statistics, the standard deviation is a measure of the amount of variation. The low standard deviation indicates that the values tend to be close to the expected value of the set, while a high standard deviation indicates that the values are spread out over a wide range.

According to the question,

The middle 50% of the pregnancies are within two standard deviations of the mean: 260 ± 2*15

               = ( 260 + 2*15 , 260 - 2*15)

               = ( 260 + 30, 260-30)

               = ( 290, 230)

Hence the range is ( 290, 230).

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Answer:cause why not

Step-by-step explanation:

Answer:

t=180-2x46=88 degrees

u=1/2(180-114)=33 degrees

The population means may be the same

So, option D is correct.

What is meant by confidence interval?

In statistics, the probability that a population parameter will fall between a set of values a predefined percentage of the time is known as the confidence interval. Analysts typically use confidence intervals that cover 95% or 99% of the expected observations while assessing data. In frequentist statistics, an estimation range for an unknown parameter is referred to as a confidence interval. The most common confidence level for computing confidence intervals is 95%, however other levels, such 90% or 99%, are also occasionally employed. The conclusion is reached that the result is statistically significant if the confidence interval eliminates the value of zero effect.

If (μ₁- μ₂=0) is included in the confidence interval, that means this two means have the same difference that is zero.

Therefore, they may be nearly same,

So, The population means may be the same

Hence, option D is correct.

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Answer:

[tex]x {}^{ \frac{19}{15} } [/tex]

Step-by-step explanation:

Hope that's helps

The measures of the angles and the values of the trigonometric functions are as follows;

33 (e) The angle is (450 - π)°

34 (f) The reference angle is 75°

40 (e) cos(θ) = -8/9

43 (e) sin(θ) = √3/2

The trigonometric functions provides the relationships between an angle and the three sides of a right triangle.

33 (e) The measure of the angle from the curve of the angle mark is found as follows;

The angle turned = 360 + (90 - θ)

Where; θ = π

Angle = 360° + (90 - π)° = (450 - π)°

34 (f) The reference angle is smallest (acute) angle made by the terminal side and the x-axis.

From the diagram, the reference angle is 360° - 285° = 75°

40 (e) The angle is in the Quadrant III

cos(θ) in Quadrant III is -cos(φ), where;

φ = θ - 180

From the diagram, cos(φ) = 8/√((-8)² + 17)

cos(θ) = -8/(√((-8)² + 17)) = -8/9

40 (f) The angle is in Quadrant II

sin(θ) = sin(φ), where; φ = 180 - θ

Therefore;

sin(φ) = 5/√((-3)² + 5²) = 5/√(34)

sin(θ) = sin(φ) = 5/√(34)

43 (e) sin(120°) = sin(180° - 120°) = sin(60°) = √3/2

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The correct option is; 4: this contradicts the Mean Value Theorem since there exists a c on (1, 7) such that f '(c) = f(7) − f(1) (7 − 1) , but f is not continuous at x = 3.

The function being used is;

f(x) = (x - 3)⁻²

If we separate this function according to x, we obtain;

f'(x) = -2/(x - 3)³

Finding all c values f(7) − f(1) = f '(c)(7 − 1).is our goal.

This suggests that;

0.06 - 0.25 = -2/(c - 3)³ x 6

-0.19 =  -12/(c - 3)³

(c - 3)³ = 63.157

c = 6.98

If the Mean Value Theorem holds for this function, then f must be continuous on [1,7] and differentiable on (1,7).

But when x = 3, f is not continuous, hence the Mean Value Theorem's prediction is false.

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The complete question is-

Let f(x) = (x − 3)−2. Find all values of c in (1, 7) such that f(7) − f(1) = f '(c)(7 − 1). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) c = Based off of this information, what conclusions can be made about the Mean Value Theorem?

Answer:$785

Step-by-step explanation:

since she took money out already you have to add the to the pre existing total.

Answer:

Your answer is B

Step-by-step explanation:

[tex]\frac{1}{4}x-5 < -6\\\\\frac{1}{4}x < -1 \\\\[/tex]

Multiply 4 to both sides leaving x by itself.

x < -4

Remember when you divide or multiply a negative to solve for x you switch the signs of greater than to less than or vice versa.

The measure of the first angle is 70° and the measure of the second angle is 100°.

The measure of complementary angles is the sum of their angles equal to 180°. Thus, if one angle has a measure of 70°, then the other angle must have a measure of 180° - 70° = 110°. Since the difference between the two angles is 14°, the first angle must have a measure of 70°, and the second angle must have a measure of 100°. This is because if the first angle has a measure of 70°, then the second angle must have a measure of 110°. Since the difference between the two angles is 14°, the second angle must have a measure of 100° in order to meet this difference. Therefore, the measure of the first angle is 70° and the measure of the second angle is 100°.

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If a lawyer charges 120 an hour then the bill for 3 hours and 20 minutes will be 400

Bill: A bill is the printed or written statement of the money owed for goods or services.In later Latin, bulla became billa, and in English billa became bill. The word can still refer to various official documents, such as a proposed law that is brought before parliament, although it is now most commonly used for documents that request payment of money.

One hour = 60 minutes

3 hours and 20 minutes = 3×60 +20

                                        = 180+20=200

Here, one hour=60 minutes charges 120

1 minute charges = 120/60

                             = 2

200 minute charges = 2×200=400

Therefore If a lawyer charges 120 an hour then the bill for 3 hours and 20 minutes will be 400.

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Answer:

x-intercept is (10,0)

y-intercept is (0, -8)

Step-by-step explanation:

Our equation is 4x - 5y = 40

Let's find x-intercepts first

X-intercepts is when the y = 0

So we put 0 in for the y, and we have the equation

4x =40

x =10

So our x-intercept located at (10,0)

Now find the y-intercepts

Y-intercepts is when the x = 10

So we put 0 in for the x, and we have the equation

-5y = 40

y= -8

So our y-intercept is located at (0,-8)

Answer:

Step-by-step explanation:

Y-int: (0, -8)

X-Int: (10, 0)

To find the y and x intercepts you have to plug in the 0 for the variable. You'd get 4(0)-5y=40 you get this by plugging in the 0 for x and then you solve. 4 x 0 is 0 so you'd get -5y=40 divide -5 on both sides and get y=-8 meaning that is the y-intercept.

You'd do the same when you look for the x-int. Just that you'd plug in a 0 for y instead of x. Which is 4x-5(0)=40. -5 x 0 is 0. So you'd have 4x=40 divide both sides by 4 and you'd get x=10 meaning that is the x intercept.

Remember that when looking for the y- intercept there should always be a zero in front meaning the x-value, and when looking for the x-intercept there should always be a  zero in the back meaning the y-value.

(Hopefully you were able to understand how I got the x and y intercepts for this equation)

The length, b, of the longest side of the triangle is 13.17 cm

From the question, we are to determine the length, b, of the longest side of the triangle

From the given information,

Area of the triangle = 22.4 cm²

Then, we can write that

22.4 = 1/2 × a × 3.7 sin97

44.8 = a × 3.7 sin97

44.8 = a × 3.6724

a = 44.8/3.6724

a = 12.199

a = 12.2

Given that the sides of the triangle are a, b, and c,

Then,

From the law of cosines, we can write that

b² = c² + a² - 2ac cosB

b² = 3.7² + a² - 2×a×3.7 cos97°

b² = 3.7² + 12.2² - 2×12.2×3.7 cos97°

b² = 3.7² + 12.2² - 2×12.2×3.7 cos97°

b² = 173.53

b = √173.53

b = 13.17 cm

The length of b is 13.17 cm

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The probability that the distance between the two points is more than 1/3 is 0.

The length of the given line segment is 1, so the two points chosen independently at random cannot be more than 1 unit apart.

That is, the maximum distance between the two points is 1. However, the question states that we are looking for the probability that the distance between them is more than 1/3.

Since the maximum distance between the two points is 1, and 1/3 is less than 1, it is impossible for the distance between them to be more than 1/3. Therefore, the probability that the distance between the two points is more than 1/3 is 0.

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Since the function increases; stops; and decreases x= -2 has to be a maximum. Similarly, x=0 is a minimum and x=2 is an inflection point. (They are "relative" max/mins as opposed to absolute.

"Critical Points" (maximums, minimum and inflection points) occur when the first derivative of the function is zero. Here f(x) is neither increasing nor decreasing at that exact point. When f '(x) is positive, f(x) is increasing; when f '(x)is negative f(x) is decreasing. In this case, the critical points are x= -2, 0, 2. For x<-2 the derivative is positive so the function is increasing, for x>-2 the derivative is negative so the function is decreasing.

b) When f '(x) is positive, f(x) is increasing; when f '(x) is negative f(x) is decreasing. In this case, the critical points are x= -2, 0, 2. For x<-2 the derivative is positive so the function is increasing, for x>-2 the derivative is negative so the function is decreasing.

Concavity is determined by the second derivative, which is easy to sketch from the first. Where f '(x) is decreasing, f ''(x)is negative; where f '(x) is increasing f ''(x) is positive. Where f '(x) is zero, f ''(x) has critical points. Now from this graph, where f ''(x) is positive, f(x) is concave up, where f ''(x)is negative, f(x) is concave down. I get that f(x) is concave up on(-1,1) and (2,3).

c) Now it should be easy to graph f from the maxs mins and critical points. There is my attempt.

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Using logical reasoning, we know that Aha's current body temperature after taking the medicine is 34°C.

Deductive reasoning, often known as logical reasoning, is the process of drawing conclusions from a body of facts or evidence.

It involves putting specific premises in front of conclusions.

Induction and abduction are two additional categories of logical reasoning that are typically distinguished in addition to the formal deduction.

If one has a premise or precondition, a conclusion or logical consequence, and a rule or material conditional that signals the conclusion given the premise, one can explain the following.

So, we know that:

The normal body temperature of the body is 37°C.

Aha current body temperature is 3 more than normal body temperature, then it is: 37 + 3 = 40°C (Current body temperature)

After taking medicine,

Temperature is 6° less than current body temperature, that is:

40 - 6 = 34°C

Temperature is 3° less than the normal body temperature, that is:
37 - 3 = 34°C

Then, the current temperature after taking medicine: 34°C

Therefore, using logical reasoning, we know that Aha's current body temperature after taking the medicine is 34°C.

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The product of the monomial and the polynomial given is 12x⁶y³-28x⁴y⁵+8x⁴y⁴-4x³y⁴

A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations.

Given that, a polynomial and a monomial 4x³y³ and (3x³-7xy²+2xy-y)

To find the product of the monomial and the polynomial, we will multiply them,

4x³y³×(3x³-7xy²+2xy-y)

= 12x⁶y³-28x⁴y⁵+8x⁴y⁴-4x³y⁴

Hence, The product of the monomial and the polynomial given is 12x⁶y³-28x⁴y⁵+8x⁴y⁴-4x³y⁴

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15 quarts are required to plan ice cream party for 60 students which is

[tex]\frac{1}{4}[/tex] th fraction of quart.

1 quarts = 4 cups

so 1 cup = [tex]\frac{1}{4}[/tex] th fraction of quart

so in  order to plan party for 60 students we need 60 cups as it is mentioned that each student gets 1 cup of ice cream.

and 1 quarts = 4 cups

so let we need x quarts for the planning of ice cream party

so total cups = 4 *x

so 4*x =60  , x= 60/4 , x=15

so 15 quarts are required for the ice cream party of 60 students  .

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The probability that he or she actually used steroids is 0.779.

The test is accurate 95% of the time when an athlete has taken steroids. It is 97% accurate when an athlete hasn't taken steroids. Suppose that the drug test will be used in a population of athletes in which 10% have actually taken steroids. Let's choose an athlete at random and administer the drug test.

Based on prior knowledge of conditions that might be related to the event, Bayes' theorem, which is named after Thomas Bayes and is used in probability theory and statistics, describes the probability of an event.

Using Baye's theorem:

P(steroids | positive) = P(steroids)*P(positive|steroids) / P(positive) = (0.10*0.95)/0.122 = 0.779

If the selected athlete passes the test, the probability that they used steroids: 0.779

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(complete question)

A company has developed a drug test to detect steroid use by athletes. The test is accurate 95% of the time when an athlete has taken steroids. It is 97% accurate when an athlete hasn’t taken steroids. Suppose that the drug test will be used in a population of athletes in which 10% have taken steroids. Let’s choose an athlete at random and administer the drug test.

Suppose that the chosen athlete tests positive. What’s the probability that he or she used steroids?

The price per quart of disinfectant is = $2.97

According to the information given in the question,

Quantity of the container of the disinfectants = 2 gallon

The price of a 2-gallon container of disinfectants = is $23.76

The concept used in this question is Ratio and Proportion,

The ratio is defined as the representation of two numbers in a fraction but in the case of ratio and proportion two numbers are separated by using a sign = ":"

Let us assume two numbers a and b,

If we have to present these two numbers in fractions = [tex]\frac{a}{b}[/tex].

If we have to present these two numbers in ratios = a : b.

According to the given question,

A 2-gallon container of disinfectants costs = $23.76

1 gallon = 4 quarts

So if 2 gallons = $23.762

= 2 gallons = 2 x 4

= 8 quarts

So, 8 quarts = $23.76

Now we will be dividing both sides by 8,

⇒ 1 quart = [tex]\frac{23.76}{8}[/tex]

⇒ 1 quart = $2.97

Therefore, the price per quart of disinfectant is = $2.97

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