solve the system of equations algebraically.

The equation of the tangent line to the curve is

y = -9 ( √3 / 2 )x + [ ( 3π√3 + 9 ) / 2 ]

What is an Equation of a line?

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

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

Given data ,

Let the curve be y = 9 cos ( x )

To find the equation of the tangent line to the curve y , take the derivative of y , we get

dy / dx = 9 ( - sin ( x ) )

dy / dx = -9 sin ( x )

So , y' = -9 sin ( x )

when x = π/3

Now plug in your value for x into y'

-9 sin ( π/3 ) = -9 ( √3 / 2 )

This is the slope of the tangent line at x = π/3

And , To find the equation of the tangent line, we need a value for y. Simply plug your x value into the original equation for y

So ,

y = 9 cos x

y = 9 cos ( π / 3 )

y = 9 ( 1/2 )

y = 9/2

Now use point slope form to find the equation of the tangent line

y - y₁ = m ( x - x₁ )

Substituting the values in the equations , we get

y - 9/2 = -9 ( √3 / 2 ) ( x - π/3 )

On simplifying the equation , we get

y - 9/2 = -9 ( √3 / 2 )x + 9 ( √3 / 2 ) ( π/3 )

Adding 9/2 on both sides of the equation , we get

y = -9 ( √3 / 2 )x + [ ( 3π√3 + 9 ) / 2 ]

Hence , the equation is y = -9 ( √3 / 2 )x + [ ( 3π√3 + 9 ) / 2 ]

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Final answer:

Bernie and Chloe hiked a total of 9 3/8 miles after completing the round trip on Tremont Trail and the Wildflower Trail.

Explanation:

Bernie and Chloe hiked the Tremont Trail which was 3 1/2 miles to the end and back, meaning they hiked 7 miles on this trail (since they had to return). After that, they hiked the Wildflower Trail, which was 2 3/8 miles long. To find out the total distance they hiked, you would add these two distances together.

The total distance for the Tremont Trail (to the end and back):
3 1/2 miles to the end + 3 1/2 miles back = 7 miles

The total distance for the Wildflower Trail:
2 3/8 miles

Adding both distances gives us:
7 miles (Tremont Trail) + 2 3/8 miles (Wildflower Trail) = 9 3/8 miles

Therefore, Bernie and Chloe hiked a total of 9 3/8 miles.

the answer is 3/10 because since they are both tenths its like doing 9 - 6

Answer:(4,-3)

Step-by-step explanation:

Answer:

37.01

Step-by-step explanation:

The solution to the system of inequalities y > x and y < -x is a V-shaped region in the second and fourth quadrants of the coordinate plane.

Graph attached.

To solve the system of inequalities graphically, you need to shade the regions on the coordinate plane that satisfy both inequalities.

The first inequality is y > x. This means that all points above the line y = x satisfy this inequality.

The second inequality is y < -x. This means that all points below the line y = -x satisfy this inequality.

Now, let's graph both lines and shade the regions that satisfy both inequalities:

Graph the line y = x. This is a straight line with a slope of 1 that passes through the origin (0,0) and goes up and to the right.

Graph the line y = -x. This is also a straight line with a slope of -1 that passes through the origin (0,0) and goes down and to the right.

Graph attached.

The shaded region that satisfies both inequalities is the area between these two lines, below y = -x, and above y = x.

It forms a V-shaped region in the second and fourth quadrants of the coordinate plane.

Therefore, the correct graph should show the shaded V-shaped region that represents the solution to the system of inequalities.

Graph attached.

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Final answer:

To write an equivalent expression using the distributive property for the expression 2(11 + y), distribute 2 to both terms inside the parentheses and simplify.

Explanation:

To write an equivalent expression using the distributive property for the expression 2(11 + y), we can distribute the 2 to both terms inside the parentheses.

This gives us:

2 * 11 + 2 * y = 22 + 2y

So, the equivalent expression using the distributive property is 22 + 2y.

2 is prime because it is a positive integer only divisible by itself and 1.

2 is the smallest prime.

A composite number is one that has more than one prime factor. 4 = 2·2 is composite, for example.

The number 1 is neither prime nor composite.

Answer:

prime

Step-by-step explanation:

Answer:

Step-by-step explanation:

[tex](6p+2)^2\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\=(6p)^2+2(6p)(2)+2^2=36p^2+24p+4\\\\\\36p^2+24p+4=36p^2+Bp+4\\\\\therefore B=24[/tex]

Answer:

The required equation is [tex]y=-4x+60[/tex]  and graph of this equation is shown below.

Step-by-step explanation:

Let y be the number of bales in the barn after feeding his cows for x days.

It is given that a farmer has 60 bales of hay stored in his barn.

Initial number of bales = 60

He feeds his cows 4 bales each day. It means number of bales decrease by 4 bales per day.

Slope = -4

The slope intercept form of a line is

[tex]y=mx+b[/tex]

where, m is slope and b is y-intercept or initial value.

Substitute m=-4 and b=60 in the above equation.

[tex]y=-4x+60[/tex]

The y-intercept of the graph is 60.

The table of values is shown below

   x               y

   0             60

  15              0

Plot these these points on coordinate plane and connect them by a straight line.

The required graph is shown below.

Answer:  The answer is (A) and (C).

Step-by-step explanation:  We are given four options and we are to check which of these inequalities have no solution.

(A) We have

[tex]x>x\\\\\Rightarrow x-x>0\\\\\Rightarrow 0>1,[/tex]

which cannot be possible, so this inequality will have no solution.

(B) We have

[tex]-3x=-3x\\\\\Rightarrow -3x+3x=0\\\\\Rightarrow 0=0,[/tex]

which is always true, so this equation will have infinitely many solutions.

(C) We have

[tex]-4+x>-2+x\\\\\Rightarrow -4+x+2-x>0\\\\\Rightarrow -2>0,[/tex]

which is never true, so this inequality will have no solution.

(D) We have

[tex]x-2<x+3\\\\\Rightarrow x-2-x-3<0\\\\\Rightarrow -5<0,[/tex]

which is always true, so this inequality will have infinitely many solutions.

Thus, the correct options are (A) and (C).

Answer:

A. x > x

C. –4 + x > –2 + x

Step-by-step explanation:

To find the distance from the point (1, -7, 8) to the plane 3x + 2y + 6z = 5, we use the formula for the distance of a point from a plane, resulting in a distance of 32/7 units.

The question deals with finding the distance from a point to a given plane. Specifically, it asks for the distance from the point (1, −7, 8) to the plane defined by the equation 3x + 2y + 6z = 5. To solve this, we use the formula for the distance of a point (x1, y1, z1) from a plane ax + by + cz = d:

D = |ax1 + by1 + cz1 − d| / √(a2 + b2 + c2)

Substituting the given values, we get:

D = |3(1) + 2(−7) + 6(8) − 5| / √(32 + 22 + 62)

This simplifies to:

D = |3 − 14 + 48 − 5| / √(9 + 4 + 36)

D = |32| / √49

D = 32 / 7

Therefore, the perpendicular distance from the point to the plane is 32/7 units.

After creating equations based on the given costs and solving for the price of chairs, desks, and beds, we find that the total cost for 5 desks and 4 beds is $725.78.

To solve this problem, we will start by setting up two equations based on the information given about the prices of chairs, desks, and beds. Let's denote the cost of one chair as C, the cost of one desk as D, and the cost of one bed as B. According to the problem, the desk costs 2 times as much as the chair, so we have D = 2C.

Now, we can create equations based on the total costs provided:

Substitute the value of D from D = 2C into the first equation:

3C + 4(2C) = 367.95 -> 11C = 367.95 -> C = 33.45

With the cost of a chair found, we can find the cost of a desk:

D = 2C = 2(33.45) = 66.90

Now, to find the cost of a bed, we use the second equation and substitute values for C:

4C + 6B = 720.72 -> 4(33.45) + 6B = 720.72 -> 133.80 + 6B = 720.72 -> 6B = 586.92 -> B = 97.82

Finally, we calculate the total cost of 5 desks and 4 beds:

5D + 4B = 5(66.90) + 4(97.82) = 334.50 + 391.28 = 725.78

The total cost of 5 desks and 4 beds is $725.78.

The negative sign indicates a decrease in the number of members. Therefore, the percentage change of -25% means that the number of members decreased by 25%. To summarize, the percentage change in the number of members from last year to this year is 25% decrease.

To find the percentage change in the number of members from last year to this year, we'll use the formula for percentage change:

[tex]\[ \text{Percentage change} = \left( \frac{\text{New value} - \text{Old value}}{\text{Old value}} \right) \times 100\% \][/tex]

Given that last year the chess club had 20 members and this year it has 15 members, we can plug these values into the formula:

[tex]\[ \text{Percentage change} = \left( \frac{15 - 20}{20} \right) \times 100\% \][/tex]

[tex]\[ \text{Percentage change} = \left( \frac{-5}{20} \right) \times 100\% \][/tex]

[tex]\[ \text{Percentage change} = -\frac{1}{4} \times 100\% \][/tex]

[tex]\[ \text{Percentage change} = -25\% \][/tex]

The negative sign indicates a decrease in the number of members. Therefore, the percentage change of -25% means that the number of members decreased by 25%.

To summarize, the percentage change in the number of members from last year to this year is 25% decrease.

The complete question is:

Last year the chess club had 20 members. This year the club has 15 members. Find the percentage of change and state whether the percent of change is an increase or decrease.

Answer:

111

Step-by-step explanation:

Answer:

111

Step-by-step explanation:

The question involves multiplying two polynomials, (x^2+3) and (x^3-5), using the distributive property to find the product x^5 - 5x^2 + 3x^3 - 15.

The question asks for the product of the polynomials (x2+3) and (x3-5). To find the product, each term in the first polynomial is multiplied by each term in the second polynomial. This is known as the distributive property or the FOIL method when dealing with binomials. Therefore, the product is:

x^5 - 5x^2 + 3x^3 - 15.

To summarize the multiplication process:

Multiply x2 by x3 to get x5

Multiply x2 by -5 to get -5x2

Multiply 3 by x3 to get 3x3

Multiply 3 by -5 to get -15

Hence, This is the standard process for multiplying polynomials.

Answer:

A significance level of 10% i.e.0.10 represents the chance of rejecting the null hypothesis

Step-by-step explanation:

Given :A hypothesis test has a significance level of 10%.

To Find :Explain what this significance level represents.

Solution :

Definition : The significance level is the probability of rejecting the null hypothesis when it is true.

So, significance level = 10%.

This means there is a 0.10 chance of rejecting the null hypothesis

Hence a significance level of 10% i.e.0.10 represents the chance of rejecting the null hypothesis