Answer
The windows do require the shutters since they cannot withstand the force of the winds.
Work Step by Step
The exercise explains that the FORCE of the wind "varies jointly" with the $area$ of the window and the $square$ of the wind's $speed$. For jointly varying variables, we can write the following equation:
$$Force_{wind} = k(Area_{window})(Speed_{wind})^{2}$$
where $k$ represents the variation coefficient. As such, we can use the information given about the first window to determine the value of $k$:
$$Force_{wind} = k(Area_{window})(Speed_{wind})^{2}$$
$$(150 lbs) = k(5 \times 4 sq ft)(30 mph)^{2}$$
$$150 = k(20)(900) = 18,000k$$
$$\frac{150}{18,000} = \frac{1}{120} = k$$
Having this value, we can re-write our initial equation in the following manner:
$$Force_{wind} = \frac{1}{120}(Area_{window})(Speed_{wind})^{2}$$
Finally, we can determine if the windows in the second part of the exercise can survive the storm by using our new equation to calculate the amount of force the windows will receive:
$$Force_{wind} = \frac{1}{120}(3 \times 4 sq ft)(60 mph)^{2}$$
$$Force_{wind} = \frac{1}{120}(12)(3,600)$$
$$Force_{wind} = 360 lbs$$
Since the windows are designed to only withstand $300$ $lbs$ of force, we can conclude that they will definitely need the shutters.
