Zach’s car travels 21 miles on 1 gallon of gas. Write an equation to represent the relationship between the gas Zach’s car uses and the distance he travels. Then solve the equation to see how far Zach travels on a trip if he uses 16 gallons of gas

formula for the tangent line is y=x-1 and differentiation of ln(x) is 1/x

A straight line without crossing the curve , it touches the curve at one single point is called tangent . Finding the derivative of a function or a variable in mathematics is called differentiation.

The ratio of the vertical to the horizontal distance between any two points on it  is called slope of ray or line or line segment in analytical geometry.

y-y1 = m(x-x1)

If the function passes through (1,0)  and the slope equals to 1 is:

y-0 = 1(x-1)

y=x-1

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Tangents are straight lines that only touch curves at a single spot rather than crossing them. Differentiation in mathematics refers to the process of determining a function's or variable's derivative.

The slope of a ray, line, or line segment in analytical geometry is the ratio of the vertical to the horizontal distance between any two locations on it.

y-y1 = m(x-x1) (x-x1)

If a function goes through the point (1, 0) and its slope is 1, then:

y-0 = 1(x-1) (x-1)

y=x-1

Among all the given equations, the one that  shows the best estimate for 4,310 ÷ 72 using compatible numbers is 4,200 ÷ 70 = 60. Hence, Option B is correct.

What are compatible numbers?

Numbers that are simple to mentally add, subtract, multiply, or divide are known as compatible numbers in mathematics. Compatible numbers are close in value to the real numbers, which facilitates estimation of the solution and simplifies problem-solving.

Compatible numbers are pairwise comparisons of numbers that are simple to mentally add, subtract, multiply, or divide. Replace real values with equivalent ones when using estimation to approximate a calculation.

When the given equation that is  4,310 ÷ 72 will solved, the derived value will be 59.86 and the selected equation will resulted in 60.

Therefore, Option B is correct.

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Addition of mixed numbers with different denominators includes converting each mixed number to an improper fraction, addition or subtraction, and simplification to a mixed number.

Here are ten examples:

3 1/4 + 2 3/5 = (3 x 4 + 1) / 4 + (2 x 5 + 3) / 5 = 13/4 + 13/5

4 2/3 + 2 1/4 = (4 x 3 + 2) / 3 + (2 x 4 + 1) / 4 = 14/3 + 9/4

5 1/2 - 3 1/3 = (5 x 2 + 1 ) / 2 - (3 x 3 + 1) / 3 = 11/2 - 10/3

3 3/4 - 1 2/5 = (3 x 4 + 3) / 4 - (1 x 5 + 2) / 5 = 15/4 - 7/5

2 1/3 + 3 2/5 = (2 x 3 + 1) / 3 + (3 x 5 + 2) / 5 = 7/3 + 17/5

Addition of mixed numbers with different denominators includes converting each mixed number to an improper fraction, finding a common denominator, converting to an equivalent fraction. addition or subtraction, and simplification to a mixed number.

additive mixed denominator unequal numbers involves the following steps:

Converts each mixed number to an improper fraction.

Finds the common denominator that is the lowest common multiple of the denominators of two or more improper fractions.

Converts each improper fraction to an equivalent fraction with a common denominator.

Add or subtract the appropriate fraction.

Simplify the result to mixed numbers if necessary.

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Rounding to the nearest tenth of an ounce, the estimated weight of a bluegill with a length of 9.5 inches is 11.8 ounces.

The goal of linear regression is to find the line of best fit, which is a straight line that minimizes the sum of the squared differences between the observed and predicted values. This line can be used to make predictions about the dependent variable based on the independent variable(s). The line is represented by an equation, called the regression equation, in which the dependent variable is a linear combination of the independent variable(s) and a constant term. In simple linear regression, there is only one independent variable, whereas in multiple linear regression, there can be two or more independent variables.

The weight of a bluegill can be estimated using a linear regression model based on the given length and weight data. To do this, we need to first find the equation of the line of best fit for the data.

The equation for a line in the form y = mx + b, where m is the slope and b is the y-intercept, can be found using the method of least squares. The slope can be calculated using the formula:

m = Σ(x - x')(y - y') / Σ(x - x')²

where x' and y' are the sample means of the length and weight data, respectively. The y-intercept can be found using the formula:

b = y' - mx'

Once we have the equation of the line of best fit, we can plug in the length of 9.5 inches to estimate the weight of a bluegill with that length.

Using the data in the table, the estimated weight of a bluegill with a length of 9.5 inches can be calculated as follows:

m = (41)(0.8) / (24) = 0.895833

b = 10.5 - (0.895833)(10) = 3.04167

y = (0.895833)(9.5) + 3.04167 = 11.80208 ounces

Rounding to the nearest tenth of an ounce, the estimated weight of a bluegill with a length of 9.5 inches is 11.8 ounces.

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The image coordinates for A, B, C are:

A' = (1, -7)

B' = (-7, 5)

C' = (-11, -4)

A translation refers to a transformation in which an object is moved a certain distance in a certain direction without changing its size or orientation. The effect of a translation is to shift the position of the object, while leaving its shape unchanged.

Since the translation rule is (x, y) => (x-1, y-3). This means you will subtract 1 from x and subtract 3 from y to get the image coordinates for A, B, C. That is:

A = (2, -4) => A' =  (2-1, -4-3) = (1, -7)

B = (-6, 8) => B' =  (-6-1, 8-3) = (-7, 5)

C = (-10, -1) => C' =  (-10-1, -1-3) = (-11, -4)

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Answer:

8890.83

Step-by-step explanation:

trust me this the answer

The limit law used to justify the computation is limₓ⇒ 2[4f(x)] = by constant multiple law of limit, So correct option is A.

In mathematics, a limit is a value that a function approaches as the input approaches a particular value. Limits are used to describe the behavior of functions near certain points and to find the derivative and integral of functions.

The formal definition of a limit is as follows: Let f(x) be a function and c be a real number. The limit of f(x) as x approaches c is L, denoted by

lim x -> c f(x) = L

if for every ε > 0, there exists a δ > 0 such that for all x, 0 < |x - c| < δ implies |f(x) - L| < ε. In other words, as x gets arbitrarily close to c, the values of f(x) get arbitrarily close to L.

The concept of limits is fundamental to calculus and is used in many applications, such as finding the maximum and minimum values of a function, determining the continuity and differentiability of a function, and solving problems in physics, engineering, and economics.

The constant multiple law of limit states that if f(x) approaches L as x approaches c, then k × f(x) approaches k × L as x approaches c, where k is a constant.

In this case, f(x) approaches 6 as x approaches 2, so limₓ⇒2 f(x) = 6.

And, 4 × f(x) = 4 × 6 = 24, so limₓ⇒2 [4f(x)] = 24 by the constant multiple law of limit.

So the answer is A: limₓ⇒2 [4f(x)] = 24 by constant multiple law of limit.

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The measure of angle BCA in the given circle O is 32°.

The angles at the circumference subtended by the same arc are equal. More simply, angles in the same segment are equal.

Given that, A, B, C and D lie on the circumference of the circle.

A and C lie on a straight line segment.

B and D lie on a straight line segment.

Here, ∠ADB=∠BCA=32° (Angles in same segment are equal)

Therefore, the measure of angle BCA in the given circle O is 32°.

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The value of a from the equation  5y = x + 47 with slope 1/5 is 11.

A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.

Any two different points on a line may be used to compute the slope of any line. The ratio of "vertical change" to "horizontal change" between two separate locations on a line is calculated using the slope of a line formula.

The increase divided by the run, or the ratio of the rise to the run, is known as the line's slope. In the coordinate plane, it describes the slope of the line. Finding the slope between two separate locations and calculating the slope of a line are related tasks. In general, we require the values of any two separate coordinates of a line in order to determine its slope.

The slope of the line = 1/5

The line passes through (8,a) and (3,10)

The equation of the line is

y = mx + c

where m is slope and c is y-intercept

or , y = x/5 + c

putting (3,10) in the line

10 = 3/5 + c

⇒ c = 47/5

The equation of the line is y = x/5 + 47/5

or, 5y = x + 47

putting (8,a) in the line

5a = 8 + 47

⇒ a = 55/5 = 11

The value of a from the equation  5y = x + 47 with slope 1/5 is 11.

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The coordinates of the vertices of the given isosceles trapezoid centered in the origin, with base 2a and OR=c are,

S(-a, 0) , Z(a, 0) , T(-b, c) and W(b, c)

Completed question :

Which could the coordinates of the vertices of the following isosceles trapezoid centered in the origin, with base 2a and OR=c?

Given data :

Given an isosceles trapezoid centered in the origin, with base 2a and OR = c.

We have to find the coordinates of vertices of the given isosceles trapezoid.

Since  O is at center then , coordinates of O (0,0).

And base SZ given to be 2a .

So SO = OZ

SO + OZ = SZ ⇒ 2 (SO) = 2a ⇒SO = a

Since point S lies in second quadrant and x coordinates in second quadrant are negative, thus Coordinate of S is (-a, 0)

and Z lies in first quadrant , both x and y are positive,

So, coordinate of Z is (a, 0)

given OR = c so R lies on y axis , so R has coordinate (0,c)

Since point T lies in second quadrant and x coordinates in second quadrant are negative, thus Coordinate of T is of the form (-x, c)

also, W lies in first quadrant , both x and y are positive. Thus, coordinate of W is of the form  (x,c)

When we compare from the given options, we get possible value of x is b,

Thus, the coordinates of the vertices of the given isosceles trapezoid centered in the origin, with base 2a and OR=c are,

S(-a, 0) , Z(a, 0) , T(-b, c) and W(b, c)

Option (3) is correct.

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The Ratio Test is inconclusive for the series sum_(n=2)^infinity 6/n^3.

What is ratio ?

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other.

The Ratio Test is given by:

lim_(n->infinity) |a_n+1 / a_n| = L

If L < 1, then the series converges. If L > 1, then the series diverges. If the limit L does not exist or is equal to 1, then the test is inconclusive.

Consider the series sum_(n=2)^infinity 6/n^3. We have:

|a_n+1 / a_n| = |6 / (n + 1)^3 / (6 / n^3)| = |n^3 / (n + 1)^3| = n^3 / (n + 1)^3

As n approaches infinity, n^3 / (n + 1)^3 approaches 1, so the limit of |a_n+1 / a_n| does not exist or is equal to,

Hence, The Ratio Test is inconclusive for the series sum_(n=2)^infinity 6/n^3.

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Answer:

Step-by-step explanation:

The formula for momentum is:

p = m * v

m_arrow * v_arrow = (m_arrow + m_apple) * v_final

m_arrow * v_arrow = (0.05 kg + 1.2 kg) * 3.6 m/s

m_arrow * v_arrow = 1.25 kg * 3.6 m/s

v_arrow = p_final / m_arrow

v_arrow = (1.25 kg * 3.6 m/s) / 0.05 kg

v_arrow = 90 m/s

So the initial velocity of the arrow was 90 m/s.

Answer:

x = 3 units

Step-by-step explanation:

Given that,

length of the square = a = √(x+6)

area of the square = 3x

We know that, the formula to find the area  of a square is:

Area = a²

3x = (√(x+6))²

3x = x + 6

Subtract x from both sides.

3x - x = 6

2x = 6

Divide both sides by 2.

x = 3

The complete two way relative frequency table shown below:

A statistical table called a two-way or contingency table displays the observed quantity or frequency for two variables, with the rows denoting one category and the columns denoting the other. In this case, the gender (male or female) row category. They can opt to have a yes or no column category.

Given:

Color     Car      Truck      Total

Blue       25          16            41

Red        21           13            34

Total       46         29            75

Now, the two way relative frequency table

Blue Car:   25 /75 x 100 = 33.33%

Blue Truck: 16/75 x 100 = 21.33%

Blue total = 41 /75 x 100 = 54.66%

Red car: 21/75 x 100 = 28%

Red Truck: 13/75 x 100 = 17.33%

Red total: 34/75 x 100 = 45.33%

Total car: 46/75 x 100 = 61.33%

Total truck: 29/75 x 100 = 38.66%

Total to total = 75/75 x 100 = 100%

so, the table is

Color     Car               Truck            Total

Blue       33.33%        21.33%       54.66%

Red        28%             17.33%        45.33%

Total       61.33%        38.66%          100%

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The denominator is the length of the interval of integration be[tex]$\frac{-\cos x}{\pi/2}$[/tex], between the limits [0, π/2]

The average value of a function throughout its domain is a loose definition of the term "mean" in calculus, and particularly in multivariable calculus. The Average function adds up the values in the given range, divides that total by the number of non-zero cells in the range, and returns the average.

A function from a set X to a set Y in mathematics assigns each element of X exactly one element of Y. The set X is known as the function's domain, while the set Y is known as the function's codomain. Initially, functions were just an idealized representation of the relationship between two changing quantities.

The integral of a function divided by the duration of the interval is the definition of the average value of a function across the interval.

Let the given function be f(x) = sin x on the interval [0, π/2]

Average [tex]$=\frac{\int \sin x d x}{\pi/2}$[/tex], over the interval [0, π/2]

The denominator is the length of the interval of integration.t.

[tex]$=\frac{-\cos x}{\pi/2}$[/tex], between the limits [0, π/2]

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Point estimators have their drawbacks, such as virtually certain error, lack of capacity to reflect larger sample sizes, and inability to infer population data.

Point estimators are useful for obtaining a single value that estimates a population parameter. However, these estimators are not always accurate as it is virtually certain that the estimate will be wrong. Additionally, point estimators are not able to reflect the effects of larger sample sizes, which means that drawing inferences about a population from a larger sample is not possible. Therefore, when using point estimators, it is important to keep in mind their limitations and use other estimation methods to achieve more accurate results.

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The proof should be written in paragraph format as follows;

Since AOB forms a line segment, ∠XOB and ∠AOX are supplementary by the linear pair theorem. Using the definition of supplementary angles, m∠XOB + m∠AOX = 180°. Since it is given that ∠XOB ≅ ∠AOX, then m ∠XOB = m ∠AOX. Applying the substitution property of equality, then 2m ∠XOB = 180°. After dividing, m∠XOB = 90°.

In Mathematics, the linear pair theorem states that the measure of two angles would add up to 180° provided that they both form a linear pair. This ultimately implies that, the measure of the sum of two adjacent angles would be equal to 180° when two parallel lines are cut through by a transversal.

According to the linear pair theorem, ∠XOB and ∠AOX are supplementary because AOB forms a line segment. Therefore, we have:

m∠XOB = m∠AOX

m∠XOB + m∠AOX = 90° + 90° = 180°

By applying the substitution property of equality, we have:

2(m∠XOB) = 2(90°) = 180°.

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Answer:

I just got it right on the test

Step-by-step explanation:

C. The probability of getting a male and getting someone with blue eyes.

P(MIB) is the probability of getting both a male and someone with blue eyes. This means that it represents the combined probability of both events happening. P(BIM) is the probability of getting a male, given that someone with blue eyes has been selected. The two probabilities are not the same because they measure different events. P(MIB) is measuring the combined probability of two events happening at the same time, while P(BIM) is measuring the probability of one event given the other has already happened.

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The car is traveling with the speed of 0.25 miles/min.

For the stated question:

Distance travelled by car ; d =  5 miles

Time taken by the care: t = 20 minuts.

Using the formula for the speed:

speed = distance /time

s = d /t

s = 5 / 20

s = 0.25 miles/min

Thus, the car is traveling with the speed of 0.25 miles/min.

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the correct question is-

A car travels 5 miles in 20 minutes. How fast was it traveling (in miles/min)

Answer:

$19,987,500.00

Step-by-step explanation:

I= prt

I= 41,000 * 3.25 * 15

41,000 * 3.25 * 15

= 19,987,500

Answer:

15

Step-by-step explanation:

A linear function has the form y = mx + b, where m is the slope and b is the y-intercept.

To find the slope (m), we can use the formula:

m = (y2 - y1) / (x2 - x1)

Plugging in the given points (3,18) and (9,24):

m = (24 - 18) / (9 - 3) = 6 / 6 = 1

To find the y-intercept (b), we can use either of the given points and the slope. Let's use the point (3,18):

18 = 1 * 3 + b

b = 18 - 3 = 15

So, the linear function rule is:

y = x + 15

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The names for the radius of the circle with center F are FA, FC, EF, BF, DF

The radius is defined as a line segment joining the center of the circle or a sphere to its circumference or boundary.

Given that, circle with center at point F. Points A, B, C, D and E are on the circle. Line segments FA, FC, AFC, EF, BF, EFB, DF and AB are drawn.

Since, the radius is the distance from center to a point on the circumference,

Therefore, FA, FC, EF, BF, DF are showing the distance from center to a point on the circumference,

Hence, the names for the radius of the circle with center F are FA, FC, EF, BF, DF

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Answer:

12

Beth has read 12 books.

Step-by-step explanation:

2/3 of 18 is

2/3 × 18

= 2/3 × 18/1

= 36/3

= 12

or "cancel" the 3 and 18 so the work looks more like:

2/3 × 18

= 2/1 × 6

= 12

Or skip the fraction math and logic it out. 18 in three parts is 6books, 6books and 6books.

Beth has read 2 parts:

6books and 6books

She read 12 books.

For any probability distribution, the expected value: 1. represents the location/center of the distribution.

In Mathematics and statistics, an expected value is sometimes referred to as the long-term mean (average) of a discrete random variable, E(X) and it can be calculated by taking the weighted average of all the outcomes of the discrete random variable with respect to their probabilities.

For instance, in a game of dice, the total number of outcomes (expected values) is generally equal to six (6). This ultimately implies that, the probability of rolling each number is p(x) = 1/6.

In conclusion, the location or center of distribution is represented by the expected value in any probability distribution.

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Euler's method is used to find the solution of the first-order derivative. The solution of Euler's method can calculate by using a calculator.

What is the first-order derivative?

The direction of the function—whether it is increasing or decreasing—is revealed by the first-order derivatives. One interpretation of the first derivative or first-order derivative is an instantaneous rate of change. It can also be predicted based on the tangent line's slope.

First-order methods, such as the Euler method, have local errors that are proportional to the square of the step size and global errors that are proportional to the step size.

Step 1: Go to Mode and select SEQ on the 5th line and then press enter.

Step 2: Input step value, and initial conditions

Step 3: Put the formula u(x) = u(x-1)+ h dy/dx.

Step 4: Press 2nd + table

Step 5: From the table, you can find the solution to the equation.

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Students are now at the dance after the additional freshmen arrive 75 students

Let x be the number of freshmen and y be the total number of people attending the dance who ARE NOT freshmen.

So...initially

x  / ( x + y)  =   40     cross-multiply

x = .40x +  .40y

x - .40x  =.40y

.60x  = .40y

y  = (3/2)x

We have that once 15 more freshmen arrive.

(x + 15)  / ( x +  (3/2)x  + 15)  =  .52

(x + 15) / ( 5/2 x  + 15)  = .52     cross-multiply

x + 15  =  .52 ( 5/2x  + 15)

x + 15 =  1.3x  + 7.8

1.3x - x  =  15 - 7.8

3x  =  7.2

3x  = 72

x = 24 =  number of original freshmen

24(3/2)  = 36  = number not freshmen

The initial attendance at the dance was 24 + (3/2)24) = 60 students.

(24) / 60   = ,40  = 40%

After 15 more freshmen arrive.....there are  60 + 15  = 75 students

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For the function C(t) = {9,          0 ≤ t ≤ 3

                            C(t) = {9 + t,     3 < t ≤ 8

                            C(t) = {20         8 < t < 28

a) If you want to give their pizza a try, in order to get the best price you should buy a large pizza on date(s) February 10, February 11, February 12, and February 13.

b) The cost of large pizza on February 18th will be $17.

c) A large pizza will cost $13 on the date February 14.

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

The given function is C(t) = {9,          0 ≤ t ≤ 3

                                   C(t) = {9 + t,     3 < t ≤ 8

                                   C(t) = {20         8 < t < 28

a) According to the function above, the promotion's lowest price for a large pizza is $9.

When t = 0, t = 1, t = 2, and t = 3, this is how much the pizza will cost.

We are aware that the symbol t stands for the duration since February 10th.

Therefore, t = 0 relates to February 10, t = 1, t = 2, and t = 3 correspond to February 11, February 12, and February 13, respectively.

Therefore, February 10, February 11, February 12, and February 13 are the best dates to test the new pizza joint in order to receive the best price.

b) Eight days after February 10th, or t = 8, is February 18th.

When 3 < t ≤ 8, the function C(t) = 9 + t can be used to calculate the price of a large pizza in dollars.

At that point, C(t) = 9 + 8 = 17 dollars when t = 8.

Therefore, on February 18, a big pizza will cost $17.

c) Because the function above says that for these values of t, C(t) = 9, we know that 13 ≠ 9 and thus a big pizza cannot cost 13 dollars when 0 ≤ t ≤ 3.

Similar to the previous example, it is also known that 13 ≠ 20 indicates that a big pizza cannot cost 13 dollars when 8 < t < 28 because C(t) = 20 for these values.

As a result, it is known that the only circumstance in which a large pizza may cost 13 dollars is when t falls within the range 3t–8 for which C(t) = 9 + t.

Enter 13 as the cost into this equation and solve for t in order to determine the day on which a large pizza will cost 13 dollars -

13 = 9 + t

t = 13 - 9

t = 4

When t = 4, this indicates that a large pizza costs $13.00.

Given that t represents the number of days since February 10, it is known that t = 4 represents February 14.

Therefore, on February 14th, a big pizza will cost $13.

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Answer:

800

Step-by-step explanation:

You would actually round 772 to 800. This is because the tens digit, 7, is greater than 5.

Answer:

800

Step-by-step explanation:

because 72 is more than 50

The number is 225. This number is an example of a number whose units digit is double the tens digit and triple the hundreds digits.

This number meets the criteria of having the units digit being double the tens digit and triple the hundreds digit. To get this number, we multiplied the hundreds digit by 3 and the tens digit by 2. The hundreds digit was 2, so 2 x 3 = 6. The tens digit was 5, so 5 x 2 = 10. Adding these together gives us 6 + 10 + 5 = 21. Lastly, we multiplied the sum of 21 by 10 to get the final result of 225. This number is an example of a number whose units digit is double the tens digit and triple the hundreds digit.

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to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below.

[tex](\stackrel{x_1}{-5}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{-8}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-8}-\stackrel{y1}{4}}}{\underset{\textit{\large run}} {\underset{x_2}{5}-\underset{x_1}{(-5)}}} \implies \cfrac{-12}{5 +5} \implies \cfrac{ -12 }{ 10 } \implies - \cfrac{ 6 }{ 5 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{- \cfrac{ 6 }{ 5 }}(x-\stackrel{x_1}{(-5)}) \implies y -4 = - \cfrac{ 6 }{ 5 } ( x +5) \\\\\\ y-4=- \cfrac{ 6 }{ 5 }x-6\implies {\Large \begin{array}{llll} y=- \cfrac{ 6 }{ 5 }x-2 \end{array}}[/tex]