Question

Volume inhaled while exercising \(=\int_{0}^{4} 1.75 \sin \frac{\pi t}{2}\) dt \(=-1.75\left(\cos \frac{\pi t}{2}\right) \times \frac{2}{\pi}\) \(=\frac{7}{\pi}\) Difference \(=\frac{7}{\pi}-\frac{5.1}{\pi}\) \(=\frac{1.9}{\pi}\) Hence, B is correct Comprehension \(5:\) $\begin{aligned} x=& a(2 \cos t+\cos 2 t), \quad y=a(2 \sin t-\sin 2 t\\\ \frac{d x}{d t} &=a(-2 \sin t-2 \sin 2 t) \\\ &=-2 a[\sin t+2 \sin 2 t] \\ &=-2 a \sin t[1+2 \cos t] \\ \frac{d y}{d t} &=a[2 \cos t-2 \cos 2 t] \\ &=-2 a\left[2 \cos ^{2} t-\cos t-1\right] \\\ &=-2 a(\cos t-1)(2 \cos t+1) \end{aligned}$ $\frac{d y}{d x}=\frac{(\cos t-1)(2 \cos t+1)}{(\sin t)(1+2 \cos t)}=-\tan t / 2$ slope of normal \(=\cot t / 2 .\)

Short answer

Now, we will integrate with respect to u: $$ \frac{3.5}{\pi}\int_{0}^{2\pi} \sin u \, du = \frac{3.5}{\pi}[-\cos u]_{0}^{2\pi} $$ By applying the limits: $$ \frac{3.5}{\pi}[-\cos (2\pi) + \cos (0)] = \frac{3.5}{\pi}[1 - (-1)] = \frac{3.5}{\pi}(2) $$ So, the integral value is: $$ \int_{0}^{4} 1.75 \sin \frac{\pi t}{2} \, dt = 7 $$ #Step 2: Calculate the difference Now that we have found the value of the integral to be 7, we can calculate the difference between the two given options: $$ | 7 - 5.5 | = 1.5 $$ Therefore, the difference in the volume inhaled while exercising between the given options is 1.5 liters.

Step-by-step solution

Calculate the integral value

First, we need to evaluate the given integral: $$ \int_{0}^{4} 1.75 \sin \frac{\pi t}{2} \, dt $$ Using the substitution method, let \(u = \frac{\pi t}{2}\). Thus, \(du = \frac{\pi}{2} dt\) and: $$ \int_{0}^{4} 1.75 \sin \frac{\pi t}{2} \, dt = \frac{3.5}{\pi}\int_{0}^{2\pi} \sin u \, du $$