Question 3 (4 points) For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of \( 194^{\circ} \) Fahrenheit was taken off the stove to cool in a \( 72^{\circ} \mathrm{F} \) room. After 23 minutes, the internal temperature of the soup was \( 96^{\circ} \mathrm{F} \), how long will it take the soup models this situation too cool to \( 75^{\circ} \mathrm{F} \) ? (round to 2 decimal place) Your Answer:

The weekly revenue for a product is given by \( R(x)=290.4 x-0.033 x^{2} \), and the weekly cost is \( C(x)=12,000+145.2 x-0.066 x^{2}+0.00002 x^{3} \), where \( x \) is the number of units produced and sold. (a) How many units will give the maximum profit? (b) What is the maximum possible profit? (a) The number of units that will give the maximum profit is (Round to the nearest whole number as needed.) (b) The maximum possible profit is \( \$ \square \).

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=3 x, y=6, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=5 \sqrt{x}, y=5, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=\sqrt{x}, y=1, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)

Use Euler's method to find an approximation for \( y(0.5) \) in the initial value problem \( y' = \frac{dy}{dx} = 4y + x \) with \( y(0) = 1 \) and \( h = 0.1 \).