Solving for r in Terms of p and t

Oct 13, 2025 by miodeya.com 49 views

Solving for r in Terms of p and t

Introduction

In this article, we will explore the given equation $$t=\sqrt{\frac{p-r}{p+r}}$$ and solve for $$r$$ in terms of $$p$$ and $$t$$. We will also find the value of $$r$$ when $$t=3$$ and $$p=10$$. This problem involves algebraic manipulation and solving a quadratic equation.

Step 1: Square Both Sides of the Equation

To eliminate the square root, we will square both sides of the equation.

t^2 = \left(\sqrt{\frac{p-r}{p+r}}\right)^2

t^2 = \frac{p-r}{p+r}

Step 2: Multiply Both Sides by (p+r)

To get rid of the fraction, we will multiply both sides of the equation by $$(p+r)$$.

t^2(p+r) = p-r

Step 3: Expand the Left Side of the Equation

We will expand the left side of the equation by multiplying $$t^2$$ by $$(p+r)$$.

t^2p + t^2r = p - r

Step 4: Move All Terms Involving r to One Side

We will move all terms involving $$r$$ to one side of the equation.

t^2p + t^2r + r = p

r(t^2 + 1) = p - t^2p

Step 5: Factor Out r

We will factor out $$r$$ from the left side of the equation.

r(t^2 + 1) = p(1 - t^2)

Step 6: Solve for r

We will solve for $$r$$ by dividing both sides of the equation by $$(t^2 + 1)$$.

r = \frac{p(1 - t^2)}{t^2 + 1}

Conclusion

We have successfully solved for $$r$$ in terms of $$p$$ and $$t$$. The value of $$r$$ is given by the equation $$r = \frac{p(1 - t^2)}{t2 + 1}$$.

Finding the Value of r When t=3 and p=10

Now that we have the equation for $$r$$ in terms of $$p$$ and $$t$$, we can find the value of $$r$$ when $$t=3$$ and $$p=10$$.

r = \frac{p(1 - t^2)}{t^2 + 1}

r = \frac{10(1 - 3^2)}{3^2 + 1}

r = \frac{10(1 - 9)}{10}

r = \frac{10(-8)}{10}

r = -8

Conclusion

We have successfully found the value of $$r$$ when $$t=3$$ and $$p=10$$. The value of $$r$$ is $$-8$$.

Final Answer

The final answer is $$r = \frac{p(1 - t^2)}{t2 + 1}$$ and $$r = -8$$ when $$t=3$$ and $$p=10$$.
Q&A: Solving for r in Terms of p and t

Introduction

In our previous article, we solved for $$r$$ in terms of $$p$$ and $$t$$ using the equation $$t=\sqrt{\frac{p-r}{p+r}}$$. We also found the value of $$r$$ when $$t=3$$ and $$p=10$$. In this article, we will answer some frequently asked questions related to solving for $$r$$ in terms of $$p$$ and $$t$$.

Q1: What is the equation for r in terms of p and t?

A1: The equation for $$r$$ in terms of $$p$$ and $$t$$ is given by $$r = \frac{p(1 - t^2)}{t2 + 1}$$.

Q2: How do I find the value of r when t=3 and p=10?

A2: To find the value of $$r$$ when $$t=3$$ and $$p=10$$, you can plug in the values of $$t$$ and $$p$$ into the equation for $$r$$ in terms of $$p$$ and $$t$$. This gives you $$r = \frac{10(1 - 3^2)}{32 + 1}$$, which simplifies to $$r = -8$$.

Q3: What if I have a different value of t and p? How do I find the value of r?

A3: If you have a different value of $$t$$ and $$p$$, you can plug in the values of $$t$$ and $$p$$ into the equation for $$r$$ in terms of $$p$$ and $$t$$. This will give you the value of $$r$$ for the given values of $$t$$ and $$p$$.

Q4: Can I use this equation to solve for p in terms of r and t?

A4: Yes, you can use this equation to solve for $$p$$ in terms of $$r$$ and $$t$$. To do this, you can rearrange the equation for $$r$$ in terms of $$p$$ and $$t$$ to solve for $$p$$. This gives you $$p = \frac{r(t^2 + 1)}{1 - t^2}$$.

Q5: What if I have a negative value of t? How do I find the value of r?

A5: If you have a negative value of $$t$$, you can plug in the value of $$t$$ into the equation for $$r$$ in terms of $$p$$ and $$t$$. This will give you the value of $$r$$ for the given value of $$t$$. Note that the equation for $$r$$ in terms of $$p$$ and $$t$$ is valid for all values of $$t$$, including negative values.

Conclusion

We have answered some frequently asked questions related to solving for $$r$$ in terms of $$p$$ and $$t$$. We have also provided the equation for $$r$$ in terms of $$p$$ and $$t$$, as well as the steps to find the value of $$r$$ when $$t=3$$ and $$p=10$$. We hope this article has been helpful in understanding how to solve for $$r$$ in terms of $$p$$ and $$t$$.

Final Answer

The final answer is that the equation for $$r$$ in terms of $$p$$ and $$t$$ is $$r = \frac{p(1 - t^2)}{t2 + 1}$$, and the value of $$r$$ when $$t=3$$ and $$p=10$$ is $$-8$$.