-7 over -10 simplified

(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.

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 angle which is 6 degrees more than 3 times its complement is found to be 69 degrees.

This problem is related to complementary angles in Mathematics. Complementary angles are two angles that add up to 90 degrees. If we denote one angle as x, its complement is 90 - x.

According to the question, an angle measure is 6 degrees more than 3 times its complement. So we write an equation from this: x = 3*(90 - x) + 6. By solving this equation, we can find the value of x which represents the angle.

First, distribute through the parenthesis: x = 270 - 3x + 6. Combine like terms: 4x = 276. Finally, divide both sides by 4 to solve for x: x = 69 degrees.

So, the angle is 69 degrees.

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Answer:

I checked and D isn't the right answer I tried it on the test and got it wrong but I guessed the second time and I got it right so the real answer is Dilation and rotation.

Given the conditions of the continuous function f and the equation f(X) = 6 where X = 1 and X = 4, f(3) must be greater than 6 because of the utilization of the Intermediate Value Theorem.

The question is about the function f which is continuous on the interval [1,5]. The given conditions are that the solutions of the equation f(X) = 6 are x = 1 and x = 4 and that f(2) = 8 which is greater than 6. By the Intermediate Value Theorem (IVT), since f is continuous on [1,5] and f(2) = 8 > 6 and we have that f(x) = 6 for x = 1 and x = 4, then by the IVT we should have a point f(c) = 6 for some c in the interval (2,4). Hence, it means that f(3) could be equal to 6 or be more than 6, but since we know the only solutions are x = 1 and x = 4, f(3) cannot be equal to 6 and thus f(3) must be greater than 6.

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The maximum number of project groups can be formed is 5 and the number of sophomores, juniors, and seniors within each group will be 5, 3 and 2 respectively.

To find the greatest number of groups that can be formed with the given distribution of students,

Identify the common factor: The greatest common factor of 30, 18, and 12 is 6.

Calculate groups: Divide each class by 6: 30 sophomores ÷ 6 = 5, 18 juniors ÷ 6 = 3, and 12 seniors ÷ 6 = 2. Therefore, the maximum number of groups that can be formed is 5.

The annual rate of change in the federal minimum wage rate from 1991 to 1997 is $0.15 per year.

The annual rate of change in the federal minimum wage rate from 1991 to 1997 can be found by calculating the difference between the final and initial values, and then dividing it by the number of years.

Initial value in 1991: $4.25

Final value in 1997: $5.15

Difference: $5.15 - $4.25 = $0.90

Number of years: 1997 - 1991 = 6 years

Annual rate of change: $0.90 / 6 = $0.15

Therefore, the annual rate of change in the federal minimum wage rate from 1991 to 1997 is $0.15 per year.

Answer:

d

Step-by-step explanation:

The answer is 7 apex

Answer-

The length of PR is 7 units.

Solution-

Given,

in the ΔDEF,

m∠D = 25°, m∠E = 114°, DF = 21, DE = 15

in the ΔPQR,

m∠P = 25°, m∠Q = 114°, PR = x, PQ = 5

So, ΔDEF ~ ΔPQR by AA (Angle-Angle) similarity, as

So, according to similarity

[tex]\Rightarrow \dfrac{DF}{PR}=\dfrac{DE}{PQ}=\dfrac{EF}{QR}[/tex]

Putting the values,

[tex]\Rightarrow \dfrac{21}{x}=\dfrac{15}{5}[/tex]

[tex]\Rightarrow x=\dfrac{21\times 5}{15}[/tex]

[tex]\Rightarrow x=7[/tex]

Answer:

as a decimal: 5.1

Answer:

The expression can be simplified by distributing the minus sign to each of the terms inside of the expression.

Step-by-step explanation:

The maximum length the number could have been before being rounded down to 40 is just under 40.5 since that is the point at which it would round up to 41.

The question pertains to rounding numbers to the nearest whole number. When rounding the number 40 to the nearest whole, we'd like to find the maximum length it could have been before rounding down. The process of rounding will round a number to the nearest whole number based on the decimal value. For instance, if you are rounding to the nearest whole number, a number with a decimal part of 0.5 or higher will round up, while a number with a decimal part of less than 0.5 will round down.

Given that 40 is already a whole number, this implies it could have been rounded down from a number up to one decimal less than 40.5 since any number 40.5 and above would round up to 41. Hence, the maximum possible length it could have been before rounding down is just under 40.5.

$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

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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Answer:

geometric sequence