If $t=\sqrt{\frac{p-r}{p+r}}$, Find:(a) $r$ In Terms Of $ P P P [/tex] And $t$.(b) The Value Of $r$ When $ T = 3 T=3 T = 3 [/tex] And $p=10$.

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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.

t2=(p−rp+r)2t^2 = \left(\sqrt{\frac{p-r}{p+r}}\right)^2

t2=p−rp+rt^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 $(p+r)$.

t2(p+r)=p−rt^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 $(p+r)$.

t2p+t2r=p−rt^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.

t2p+t2r+r=pt^2p + t^2r + r = p

r(t2+1)=p−t2pr(t^2 + 1) = p - t^2p

Step 5: Factor Out r

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

r(t2+1)=p(1−t2)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=p(1−t2)t2+1r = \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 $r = \frac{p(1 - t2)}{t2 + 1}$.

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=p(1−t2)t2+1r = \frac{p(1 - t^2)}{t^2 + 1}

r=10(1−32)32+1r = \frac{10(1 - 3^2)}{3^2 + 1}

r=10(1−9)10r = \frac{10(1 - 9)}{10}

r=10(−8)10r = \frac{10(-8)}{10}

r=−8r = -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 $r = \frac{p(1 - t2)}{t2 + 1}$ 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 $r = \frac{p(1 - t2)}{t2 + 1}$.

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 $r = \frac{10(1 - 32)}{32 + 1}$, 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 $r = \frac{p(1 - t2)}{t2 + 1}$, and the value of $r$ when $t=3$ and $p=10$ is $-8$.