Two trains leave the same station at the same time, but are headed in opposite directions. One train travels at 50 miles per hour and the other train travels at 60 miles per hour. How much time passes until the trains are 660 miles apart?

The value of y'' will be (- 5x/y² - 25x⁴/2y⁵).

What is implicit differentiation?

In implicit differentiation, we differentiate each side of an equation with two variables by treating one of the variable as a function of other.

Given that;

The equation is;

5x³ + 2y³ = 1

Now, Apply implicit differentiation rule we get;

5x³ + 2y³ = 1

After differentiation;

15x² + 6y² dy/dx = 0

dy/dx = - 15x² / 6y²

dy/dx = -5x²/2y²

Again differentiation;

d²y/dx² = -5/2 (y² × 2x - x² × 2y dy/dx) / y⁴

d²y/dx² = -5/2 (2xy² - 2x²y dy/dx) / y⁴

Substitute value of dy/dx;

d²y/dx² = - 5/2 (2xy² - 2x²y × -5x²/2y²) / y⁴

d²y/dx² = - 5/2 (2xy² + 5x⁴/y) / y⁴

d²y/dx² = - 5x/y² - 25x⁴/2y⁵

y'' = - 5x/y² - 25x⁴/2y⁵

Thus, The value of y'' will be (- 5x/y² - 25x⁴/2y⁵).

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1+3^2 = 1+9 = 10

10/5 = 2

2-6+2 = -4+2

 answer is -2

The equation is [tex]\(x^2 + 12y^2 - 3072 = 0\),[/tex] obtained from integrating [tex]\(12y \, dy = -x \, dx\)[/tex] with the initial condition.

To find the equation of the curve passing through the point (0, 16) with a slope given by [tex]\( \frac{{dy}}{{dx}} = -\frac{x}{12y} \),[/tex] you can use separation of variables.

Given that [tex]\( \frac{{dy}}{{dx}} = -\frac{x}{12y} \)[/tex], we can rewrite it as:

[tex]\[ 12y \, dy = -x \, dx \][/tex]

Now, integrate both sides:

[tex]\[ \int 12y \, dy = \int -x \, dx \]\[ 6y^2 = -\frac{1}{2}x^2 + C \][/tex]

Given that the curve passes through the point (0, 16), substitute these values into the equation:

[tex]\[ 6(16)^2 = -\frac{1}{2}(0)^2 + C \]\[ 6(256) = C \]\[ C = 1536 \][/tex]

Therefore, the equation of the curve is:

[tex]\[ 6y^2 = -\frac{1}{2}x^2 + 1536 \][/tex]

This can also be written in standard form as:

[tex]\[ x^2 + 12y^2 - 3072 = 0 \][/tex]

So, the equation of the curve is [tex]\( x^2 + 12y^2 - 3072 = 0 \).[/tex]

Shantel will need to study for 60 minutes more to complete her history review.

Shantel is studying history and after 20 minutes, she is [tex]\frac{1}{4}[/tex]  of the way done.

To determine how much longer she will study, we need to calculate how much time is needed to complete the remaining [tex]\frac{3}{4}[/tex] of her study time.

[tex]\frac{1}{4}[/tex] of the study done in 20 minutes means she will study for 4 * 20 = 80 minutes in total.

To find out how much longer she needs to study, subtract the time she has already studied (20 minutes) from the total study time (80 minutes): 80 - 20 = 60 minutes.

Therefore, Shantel will need to study for 60 minutes more to complete her history review.

25  Is the largest number too add up to the sum.

Final answer:

To solve the equation 2 + √(4y - 3) = 11 for y, subtract 2 from both sides, square both sides to remove the square root, then solve for y to find that y = 21.

Explanation:

The question asks to solve the equation 2 + √(4y - 3) = 11 for y. To solve for y, follow the steps below:

First, subtract 2 from both sides of the equation to isolate the square root term: √(4y - 3) = 9.

Now, square both sides of the equation to eliminate the square root: (√(4y - 3))^2 = 9^2, which simplifies to 4y - 3 = 81.

Add 3 to both sides of the equation to isolate the term with y: 4y = 84.

Finally, divide both sides by 4 to solve for y: y = 21.

Therefore, the solution to the equation is y = 21.

Solution:

we have been asked to write the number 63 in four different ways. It can be done as follows

[tex]63=60+3[/tex]

Here we are just adding 3 to 60 and finally we get 63.

[tex]63=6\times10+3[/tex]

This one of the many way of writing 63.

[tex]63=70-7[/tex]

Here we have used the concept of subtraction to write 63.

[tex]63=7\times10-7[/tex]

Concept of subtraction and multiplication to write the number 63.

Hence  the four ways have been written.

The question is about using the Monte Carlo method to estimate the value of π, by generating random points in a square, counting how many fall inside a unit circle, and finally using this count to calculate π.

This question involves the implementation of a famous problem in computational mathematics known as the Monte Carlo method for estimating the value of π:

This process is based on the mathematical fact that the ratio of the area of a unit circle to the square it is inscribed in is π/4.

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To estimate the value of sin pi/3, which is roughly 1.047 or 60 degrees, we can use standard trigonometric values. Since pi/3 is closer to pi/2 than pi, where sin achieves its maximum of 1, the value of sin pi/3 should be closer to 1 than 0. Standard trigonometric values confirm that sin pi/3 = sqrt(3)/2 = 0.866.

To estimate sin pi/3, we can make use of some standard values in trigonometry. Pi is roughly 3.14, hence pi/3 is roughly 1.047. The sin function gives a result of 1 at pi/2 (or 90 degrees), 0 at 0, and again 0 at pi (or 180 degrees). Thus, it achieves its maximum at pi/2 and then decreases to 0 at pi. Intuitively, since pi/3 (or 60 degrees) is closer to pi/2 than to pi, the sin value at pi/3 should be closer to 1 than to 0. From the standard trigonometric values, we know that sin pi/3 = sqrt(3)/2 which is roughly 0.866.

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