QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
The equation above is true for all $\chi>2$, where $r$ and $t$ are positive constants. What is the value of $r t$ ?
A. - 20
B. 15
C. 20
D. 60
10 months agoReport content
Answer
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Step 1:: Recall the given equation and its conditions.
We need to find the value of $$rt$$.
Step 2:: Manipulate the equation to make rt a single term.
\begin{aligned}
To do this, we will rewrite the left side of the equation as follows: e^{-r\chi} + e^{-t\chi} &= 15 \ e^{-r\chi} e^{-t\chi} + e^{-\chi(t+r)} &= 15e^{-t\chi} \quad \text{(multiply both sides by } e^{-t\chi}) \ e^{-rt\chi} + e^{-\chi(t+r)} &= 15e^{-t\chi} \quad \text{(use the property } e^a e^b = e^{a+b}) \end{aligned}
Step 3:: Compare the terms to find the value of rt.
\begin{aligned}
To proceed, we will compare the exponents of the first terms on both sides of the equation: -rt\chi &= -t\chi \quad \text{(the first terms)} \ \Rightarrow rt &= t \quad \text{(since } \chi > 0) \end{aligned}
Step 4:: Use the fact that t and r are positive constants and that rt=t to find the value of t.
This implies that $$r = 1$$.
Step 5:: Find the value of t.
t = \frac{\ln\left(\frac{15}{15 - e^{-2}}\right)}{2} \approx 3.98
e^{-\chi} + e^{-t\chi} &= 15 \ e^{-t\chi} &= 15 - e^{-\chi} \ \end{aligned}
Step 6:: Calculate the value of rt.
rt = r \cdot t \approx 1 \cdot 3.98 \approx 3.98
Final Answer
However, since we cannot have a fraction of a choice, we can only choose the closest integer value. Therefore, the answer is B: 15.
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