Possible Dimensions: The height is at least 38 inches, and the base side length of the mailer is at least about 10.39 inches
B
bHeight: 38 inches Side Base Length: About 10.39 inches
Practice makes perfect
a A mailer in the shape of a triangular prism needs to hold a poster that is almost 38 inches long and has a maximum rolled diameter of 6 inches. Let's choose an equilateral triangle to be a base of the prism.
Now we will find the dimensions of the prism h and s such that the poster will fit into. Since the poster is almost 38 inches long, the height of the mailer should be at least 38 inches.
Height of Mailer: h≥ 38 inches
Since a maximum rolled diameter of the poster is 6 inches, the base of the mailer should fit a circle with a diameter equal to at least 6 inches.
To find what values s can take, let's analyze the situation when s is as small as possible. That is when the circle is inscribed in the triangle.
Since the diameter of the circle is 6 inches, its radius is r= 62=3 inches. Let's sketch altitudes of the triangle to find the minimum length of s.
Notice that the altitudes of the equilateral triangle are also its medians. From the Centroid Theorem, the centroid C is two-thirds of the distance from each vertex to the midpoint of the opposite side. This tells us that BC is two times larger than AC.
BC=2* AC=2* 3=6 in.
Now we can find the height, h=AB.
Therefore, the side length of the equilateral triangle is about s=10.39 inches. This tells us that the side of the base in the mailer should be at least 10.39 inches.
Base Side Length of Mailer:
s≥ 10.39 in.
b If we want to minimize the surface area of the mailer, the dimensions of the mailer should be as small as possible. From Part A we know that the height should be at least 38 inches, and the side base length should be at least about 10.39 inches.
Height of Mailer: &h≥ 38 in.
Base Side Length of Mailer: &s≥ 10.39 in.
This tells us that we minimize the surface area of the mailer when h=38 inches and s=10.39 inches.
