A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3 additional seats . How many seats are in thecivic theater

The square root of 35 is an irrational number because it cannot be expressed as a precise fraction or whole number.

The square root of 35 is an irrational number. Let me explain why. A number is rational if it can be expressed as a fraction of two integers, in other words, if it can be written in the form a/b where a and b are integers and b ≠ 0. The square root of a perfect square (like 9, 16, 25 and so on) is a rational number because it can be expressed as a whole number. But if a number is not a perfect square (like 35), its square root is generally an irrational number. The square root of 35 cannot be expressed as a whole number or as a precise fraction, hence it is an irrational number.

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(a) The value of [tex]\frac{dy}{dt}[/tex] is [tex]\boxed{\dfrac{dy}{dt}=\dfrac{7}{8}}[/tex].

(b) The value of [tex]\frac{dx}{dt}[/tex] is [tex]\boxed{\dfrac{dx}{dt}=128}[/tex].

Further explanation:

Given that [tex]x[/tex] and [tex]y[/tex] are both differentiable functions of [tex]t[/tex].

The function is as follows:

[tex]\fbox{\begin\\\ \math y=\sqrt{x}\\\end{minispace}}[/tex]

Derivatives of parametric functions:

The relationship between variable [tex]x[/tex] and variable [tex]y[/tex] in the form [tex]y=f(t)[/tex] and [tex]x=g(t)[/tex] is called as parametric form with [tex]t[/tex] as a parameter.

The derivative of the parametric form is given as  follows:

[tex]\fbox{\begin\\\ \dfrac{dy}{dt}=\dfrac{dy}{dx}\times\dfrac{dx}{dt}\\\end{minispace}}[/tex]  

The above equation is neither explicit nor implicit, therefore a third variable is used.

Part (a):

It is given that [tex]x=16[/tex] and [tex]\frac{dx}{dt}=7[/tex].

To find the value of [tex]\frac{dy}{dt}[/tex] first find the value of [tex]\frac{dy}{dx}[/tex] that is shown below:

[tex]\begin{aligned}y&=\sqrt{x}\\\dfrac{dy}{dx}&=\dfrac{1}{2\sqrt{x}}\end{aligned}[/tex]  

The value of [tex]\frac{dy}{dt}[/tex] is calculated as  follows:

[tex]\begin{aligned}\dfrac{dy}{dt}&=\dfrac{dy}{dx}\times\dfrac{dx}{dt}\\&=\dfrac{1}{2\sqrt{x}}\dfrac{dx}{dt}\end{aligned}[/tex]

Now, substitute the value of [tex]x[/tex] and [tex]\frac{dx}{dt}[/tex] in above equation as shown below:

[tex]\begin{aligned}\dfrac{dy}{dt}&=\dfrac{1}{2\sqrt{16}}\times7\\&=\dfrac{1}{2\times4}\times7\\&=\dfrac{7}{8}\end{aligned}[/tex]  

Therefore, the required value of [tex]\frac{dy}{dt}[/tex] is [tex]\frac{7}{8}[/tex].

Part (b):

It is given that [tex]x=64[/tex] and [tex]\frac{dy}{dt}=8[/tex].

To find the value of [tex]\frac{dx}{dt}[/tex] first find the value of [tex]\frac{dx}{dt}[/tex] that is shown below:

[tex]\begin{aligned}y&=\sqrt{x}\\\dfrac{dy}{dx}&=\dfrac{1}{2\sqrt{x}}\\\dfrac{dx}{dy}&=2\sqrt{x}\end{aligned}[/tex]

The value of [tex]\frac{dx}{dt}[/tex] is calculated as  follows:

[tex]\boxed{\dfrac{dx}{dt}=\dfrac{dx}{dy}\times\dfrac{dy}{dt}}[/tex]

Now, substitute the value of [tex]\frac{dx}{dy}[/tex] and [tex]\frac{dy}{dt}[/tex] in above equation as shown below:

[tex]\begin{aligned}\dfrac{dx}{dt}&=2\sqrt{64}\times 8\\&=2\times8\times8\\&=2\times64\\&=128\end{aligned}[/tex]

Therefore, the required value of [tex]\frac{dx}{dt}[/tex] is [tex]128[/tex].

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Answer details:

Grade: Senior school

Subject: Mathematics

Chapter: Derivatives

Keywords:  dy/dt, dx/dt, y=rootx, derivatives, parametric form, implicit, explicit, function, differentiable, x, y, t,  x=16, x=64, dy/dx, dx/dy, differentiation.

The loser had 5.5 points in the chess match.

To find out how many points the loser had in the chess match, we can set up an equation. Let's say the loser had x points. The winner was 4 points ahead, so the winner had x + 4 points. Since there were 15 games in total, we can set up the equation:

x + (x + 4) = 15

Combining like terms, we get:

2x + 4 = 15

Subtracting 4 from both sides, we get:

2x = 11

Dividing both sides by 2, we get:

x = 5.5

Therefore, the loser had 5.5 points in the chess match.

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$150 per night

Use a system of equations using x (cost of hotel) and y (cost of car): 

5x + 6y = 1170 

7x + 8y = 1610 

Solve for y in one of the equations and then use substitution to solve for x in the other: 

6y = 1170 - 5x 

y = (-5/6)x + 195 

7x + 8((-5/6)x + 195) = 1610 

7x + (-20/3)x + 1560 = 1610 

(1/3)x +1560 = 1610 

(1/3)x = 50 

x = $150 a night

Answer:- A reflection of the line segment across the line y = –x .

Explanation:-

A reflection over the line y = -x, the x-coordinate and y-coordinate interchange their places and they are negated (the signs are changed).

Given :- A line segment has endpoints at (–1, 4) and (4, 1) such that it reflects produce an image with endpoints at (–4, 1) and (–1, –4).

(-1, 4)→(-4, 1) and

(4, 1)→(-1, -4)

Thus this shows a reflection of the line segment across the line y = –x.

Answer:

A reflection of the line segment across the line y = –x.

Step-by-step explanation:

Given : A line segment has endpoints at (–1, 4) and (4, 1).

To find : Which reflection will produce an image with endpoints at (–4, 1) and (–1, –4).

Solution : We have given that endpoints at (–1, 4) and (4, 1).

After reflection endpoints become  (–4, 1) and (–1, –4).

Reflection:   (–1, 4)  →→ (–4, 1)

                    (4, 1)→→→ (–1, –4).

We can see x coordinate become -y coordinate and y coordinate become -x coordinate.

(x ,y) →→ (–y, -x)

This is the a reflection of the line segment across the line y = –x rule of reflection .

Therefore, a reflection of the line segment across the line y = –x.

Answer:

x represents Annette speed in miles per minute. Hence, option b) is correct.

Step-by-step explanation:

Given that Annette rode her bike 6.8 miles at a constant rate for 90 minutes. Also to find how fast she traveled, she solved the equation [tex]6.8=x\times 90[/tex]. Now, we have to tell here what x represents.

As, [tex]Speed=\frac{Distance}{Time}[/tex]

Let speed of bike is x miles/min and given distance is 6.8 miles and at a constant rate for 90 minutes.

∴ [tex]x=\frac{6.8}{90}[/tex]

⇒  [tex]6.8=x\times 90[/tex]

which is the same equation as given.

Here x represents Annette speed in miles per minute. Hence, option b) is correct.

Answer:

distributive property (a)

Step-by-step explanation:

I got 100/100 so this should be correct for you too.

Bob's team won 12 out of the 16 games it played, which is calculated by multiplying the total number of games (16) by the percentage of games won (75%).

Bob's team won 75% of its baseball games and played 16 games in all. To find out how many games the team won, we can use the formula:

Number of games won = (Percentage of games won / 100) x Total number of games played

Number of games won = (75 / 100) x 16

Number of games won = 0.75 x 16

Number of games won = 12

So, Bob's team won 12 out of the 16 games played.

The corresponding equation is y = 1/5(x).

A graph of the equation is shown below.

In Mathematics and Geometry, a proportional relationship is a type of relationship that passes through the origin (0, 0) and produces equivalent ratios as represented by the following mathematical equation:

y = kx

Where:

y represents the y-variable​.

x represents the x-variable.

k is the constant of proportionality.

Next, we would determine the constant of proportionality (k) by using the various data points in table B as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 4/20

Constant of proportionality, k = 1/5.

In this context, the corresponding equation is given by;

y = 1/5(x)

The equivalent factored form is A. 7x(2x + 5).

To factorize the expression 14x^2 + 35x completely, we look for the greatest common factor (GCF) of the terms and apply the distributive property.

In this case, the GCF of 14x^2 and 35x is 7x.

So, we can factor out 7x from the expression:

14x^2 + 35x = 7x(2x + 5)

Therefore, the equivalent factored form is A. 7x(2x + 5).

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Answer:

Step-by-step explanation:

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The total cost to produce 30 bicycles is $5599.89. The profit from first 30 bikes is $550.11. The marginal profit on the 31st bike is $126.57.

The marginal cost function given is 600/(0.3q+5). To find the total cost to produce 30 bicycles, we need to integrate the marginal cost from 0 to 30 and add the fixed cost of 2200.

The result of the integration is 2000*(ln|1+0.3q|). Calculating from 0 to 30 gives us 3599.89. Thus, the total cost to produce 30 bicycles considering the fixed cost is 5599.89.

The revenue of each bike sold is $205. So, the total revenue generated from selling 30 bikes is 30*$205 = $6150. The profit is then the difference between the revenue and the total cost, giving us $6150-$5599.89 = $550.11.

To find the marginal profit on the 31st bike, subtract the marginal cost at 31 bikes from the selling price. The marginal cost at 31 bikes is the marginal cost function evaluated at q=31, which is approximately $78.43, and hence, the marginal profit on the 31st bike is $205 - $78.43 = $126.57.

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To find the total cost, multiply the quantity by the marginal cost function and add the fixed cost. To find the profit or loss, subtract the total cost from the revenue. The marginal profit can be found by subtracting the marginal cost from the selling price.

To find the total cost to produce 30 bicycles, you need to calculate the variable cost and add it to the fixed cost. The variable cost can be found by multiplying the marginal cost function by the quantity of bikes. So, the variable cost is (600/0.3*30 + 5) = 6105. Adding the fixed cost of 2200 gives us a total cost of 8305.

To find the profit or loss on the first 30 bicycles, we need to subtract the total cost from the revenue, which is the selling price multiplied by the quantity. So, the revenue is 205 * 30 = 6150. The profit or loss is then 6150 - 8305 = -2155, indicating a loss of $2155.

The marginal profit on the 31st bicycle can be found by subtracting the marginal cost from the selling price. So, the marginal profit is 205 - (600/0.3*31 + 5) = -55, indicating a loss on the 31st bicycle.

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Answer:

The correct option is B) -3

Step-by-step explanation:

Consider the provided expression.

[tex]5x^3 + 2[/tex]

We need to solve the expression when x=-1.

Substitute the values of x in the provided expression.

[tex]5(-1)^3 + 2[/tex]

Simplify the expression.

[tex]5\times -1+ 2[/tex]

[tex]-5+ 2[/tex]

[tex]-3[/tex]

Hence, the value of the provided expression when x=-1 is -3.

Therefore, the correct option is B) -3