One particular 13-foot tall cone of dry sand has a base diameter of 38.6 feet. To the nearest tenth of a degree, what is the angle of repose of this type of dry sand?A different conical pile of the same type of sand as before (which will have the same angle of repose) is 10 feet tall. What is the diameter of the cone’s base?Show your mathematical steps and explain your process in words.

34.0 degrees, 29.6 feet in Diameter.
This is a right triangle trigonometry question.
Use half of the diameter to form the base of the triangle (19.3 ft) with a height 0f 13.
Use Inverse Tangent function to find the angle of repose
[tex]tan ^{-1}(\frac{Opposite}{Adjacent} ) = Angle of repose [/tex]

[tex]tan ^{-1}(\frac{13}{19.3} ) = 33.96 [/tex] round to 34.0

then use the angle of repose to solve trigonometric function of the tangent for the radius of the other cone.

[tex]tan(Angle)= \frac{opp}{adj} [/tex]

[tex]tan(34)=(\frac{10}{x} ) [/tex]

switch denominator and tangent

[tex]x=(\frac{10}{tan(34)} ) =14.82[/tex]
multiply by 2 to get diameter of second cone
14.82*2≈29.6