The radii of the bases of the truncated cone are 11 and 16 cm generatrix = 13 cm

The radii of the bases of the truncated cone are 11 and 16 cm generatrix = 13 cm find the distance from the center of the smaller base to the point of the circle of the larger base.

The quadrilateral formed by two radii, the axis and the generatrix of the truncated cone, is a rectangular trapezoid of AВСD with right angles ∠C and ∠D.

To begin with, we find the length of the smaller side of the СD, which is the height of this cone.

Since the НD segment is equal to the length of the smaller base of the BC, then:

AН = AD – НD;

AH = 16 – 11 = 5 cm.

Consider the triangle ΔАВН. According to the Pythagorean theorem:

AB ^ 2 = BH ^ 2 + AH ^ 2;

BH ^ 2 = AB ^ 2 – AH ^ 2;

BH ^ 2 = 13 ^ 2 – 5 ^ 2 = 169 – 25 = 144;

BH = √144 = 12 cm;

СD = BH = 12 cm.

The distance from the center of the smaller base to the point of the circle of the larger base is the hypotenuse of the right-angled triangle ΔAСD. Therefore, we can also find its length using the Pythagorean theorem:

AC ^ 2 = СD ^ 2 + AD ^ 2;

AC ^ 2 = 16 ^ 2 + 12 ^ 2 = 256 + 144 = 400;

AC = √400 = 20 cm.

Answer: The length of the speaker is 20 cm.