💡Alright, so I have this problem where I need to enclose a sphere of radius ( r ) within both a complete cone and a truncated cone. The volume of each cone is supposed to be twice that of the sphere. I need to find the height of the complete cone, the radius of its base, the radii of the base and top faces of the truncated cone, and then determine the ratio of the tangents of their slant heights.First, let me understand the problem. I have a sphere with radius ( r ). I need to fit this sphere inside two different cones: one complete cone and one truncated cone (which is like a cone with the top cut off). Both cones have a volume that's twice the volume of the sphere. So, I need to find the dimensions of these cones and then compare their slant heights.Let me start with the complete cone. The volume of a sphere is ( frac{4}{3}pi r^3 ). So, twice that volume would be ( frac{8}{3}pi r^3 ). The volume of a cone is ( frac{1}{3}pi R^2 h ), where ( R ) is the base radius and ( h ) is the height. So, setting the volume of the cone equal to twice the sphere's volume:[frac{1}{3}pi R^2 h = frac{8}{3}pi r^3]Simplifying, I get:[R^2 h = 8 r^3]So, ( R^2 h = 8 r^3 ). That's one equation relating ( R ) and ( h ).Now, since the sphere is enclosed within the cone, it should be tangent to the cone. I need to relate the radius and height of the cone to the radius of the sphere. Let me visualize this: the sphere is inside the cone, touching the sides. So, if I draw a cross-sectional view of the cone and the sphere, it would look like a triangle with a circle inside it, tangent to the sides and the base.In this cross-section, the cone is an isoceles triangle, and the sphere is a circle inscribed in it. The center of the sphere will be along the axis of the cone. Let me denote the radius of the sphere as ( r ), the radius of the cone's base as ( R ), and the height of the cone as ( h ).From the cross-sectional view, the sphere touches the cone at some point. The distance from the center of the sphere to the apex of the cone is ( h - r ), since the sphere's center is ( r ) units above the base.Using similar triangles, I can relate the dimensions of the cone to the sphere. The triangle formed by the apex, the center of the sphere, and the point where the sphere is tangent to the cone's side is similar to the entire triangle of the cone.Let me denote the distance from the apex to the center of the sphere as ( h - r ). The radius of the sphere is ( r ), and the radius of the cone at the height of the center is something I need to find. Wait, actually, at the height ( h - r ) from the apex, the radius of the cone would be proportional.The entire cone has height ( h ) and base radius ( R ). So, the slope of the cone's side is ( frac{R}{h} ). At a height ( h - r ) from the apex, the radius would be ( frac{R}{h} times (h - r) ).But this radius should also be equal to the radius of the sphere divided by the sine of the angle between the cone's side and the axis. Wait, maybe I need to use the formula for the radius of an inscribed sphere in a cone.I recall that for a sphere inscribed in a cone, the relationship between the sphere's radius ( r ), the cone's height ( h ), and the cone's base radius ( R ) is given by:[frac{r}{h} = frac{R}{sqrt{R^2 + h^2}}]Is that correct? Let me think. The sphere is tangent to the cone's side, so the distance from the center of the sphere to the cone's side is equal to the sphere's radius. The distance from a point to a line in 3D can be calculated, but maybe in the cross-sectional view, it's simpler.In the cross-sectional view, the cone is a triangle with base ( 2R ) and height ( h ). The sphere is a circle with radius ( r ) inscribed in this triangle. The center of the circle is at a distance ( r ) from the base and at a distance ( h - r ) from the apex.The distance from the center of the circle to the side of the triangle is equal to the radius ( r ). Using the formula for the distance from a point to a line, I can set up an equation.The side of the triangle can be represented by the line ( y = frac{R}{h} x ). The center of the circle is at ( (0, r) ). The distance from this point to the line ( y = frac{R}{h} x ) is equal to ( r ).The distance from a point ( (x_0, y_0) ) to the line ( ax + by + c = 0 ) is ( frac{|ax_0 + by_0 + c|}{sqrt{a^2 + b^2}} ).So, rewriting ( y = frac{R}{h} x ) as ( frac{R}{h} x - y = 0 ), the distance from ( (0, r) ) to this line is:[frac{|frac{R}{h} cdot 0 - 1 cdot r + 0|}{sqrt{left(frac{R}{h}right)^2 + (-1)^2}} = frac{| - r |}{sqrt{frac{R^2}{h^2} + 1}} = frac{r}{sqrt{frac{R^2}{h^2} + 1}} = r]Wait, that can't be right because it would imply:[frac{r}{sqrt{frac{R^2}{h^2} + 1}} = r implies sqrt{frac{R^2}{h^2} + 1} = 1 implies frac{R^2}{h^2} + 1 = 1 implies frac{R^2}{h^2} = 0]Which would imply ( R = 0 ), which doesn't make sense. I must have made a mistake in setting up the distance formula.Wait, perhaps the center of the sphere isn't at ( (0, r) ). In the cross-sectional view, the sphere is tangent to the base of the cone, which is at ( y = 0 ), and also tangent to the sides. So, the center of the sphere should be at ( (0, r) ), because it's ( r ) units above the base.But then, the distance from the center to the side should be equal to the radius ( r ). So, maybe my mistake is in the equation of the line.Wait, the side of the cone is not ( y = frac{R}{h} x ), because in the cross-sectional view, the apex is at ( (0, h) ), and the base is at ( (R, 0) ) and ( (-R, 0) ). So, the right side of the cone is the line from ( (0, h) ) to ( (R, 0) ).So, the slope of this line is ( frac{0 - h}{R - 0} = -frac{h}{R} ). So, the equation of the line is ( y = -frac{h}{R} x + h ).So, the distance from the center ( (0, r) ) to this line should be equal to ( r ).Let me write the equation of the line in standard form:[y = -frac{h}{R} x + h implies frac{h}{R} x + y - h = 0]So, the distance from ( (0, r) ) to this line is:[frac{|frac{h}{R} cdot 0 + 1 cdot r - h|}{sqrt{left(frac{h}{R}right)^2 + 1^2}} = frac{|r - h|}{sqrt{frac{h^2}{R^2} + 1}} = frac{h - r}{sqrt{frac{h^2}{R^2} + 1}}]Since ( h > r ), the absolute value is ( h - r ).This distance should be equal to the radius ( r ):[frac{h - r}{sqrt{frac{h^2}{R^2} + 1}} = r]Let me square both sides to eliminate the square root:[frac{(h - r)^2}{frac{h^2}{R^2} + 1} = r^2]Multiply both sides by the denominator:[(h - r)^2 = r^2 left( frac{h^2}{R^2} + 1 right )]Expand the left side:[h^2 - 2 h r + r^2 = r^2 left( frac{h^2}{R^2} + 1 right )]Simplify the right side:[h^2 - 2 h r + r^2 = frac{r^2 h^2}{R^2} + r^2]Subtract ( r^2 ) from both sides:[h^2 - 2 h r = frac{r^2 h^2}{R^2}]Let me rearrange this:[h^2 - 2 h r = frac{r^2 h^2}{R^2}]Multiply both sides by ( R^2 ):[R^2 h^2 - 2 R^2 h r = r^2 h^2]Bring all terms to one side:[R^2 h^2 - 2 R^2 h r - r^2 h^2 = 0]Factor out ( h^2 ):[h^2 (R^2 - r^2) - 2 R^2 h r = 0]Hmm, this seems complicated. Maybe I should express ( R ) in terms of ( h ) from the volume equation.From the volume equation earlier:[R^2 h = 8 r^3 implies R^2 = frac{8 r^3}{h}]So, substitute ( R^2 = frac{8 r^3}{h} ) into the equation:[h^2 left( frac{8 r^3}{h} - r^2 right ) - 2 cdot frac{8 r^3}{h} cdot h r = 0]Simplify term by term:First term:[h^2 left( frac{8 r^3}{h} - r^2 right ) = h^2 cdot frac{8 r^3 - h r^2}{h} = h (8 r^3 - h r^2)]Second term:[- 2 cdot frac{8 r^3}{h} cdot h r = -2 cdot 8 r^3 cdot r = -16 r^4]So, putting it all together:[h (8 r^3 - h r^2) - 16 r^4 = 0]Expand:[8 r^3 h - h^2 r^2 - 16 r^4 = 0]Let me factor out ( r^2 ):[r^2 (8 r h - h^2 - 16 r^2) = 0]Since ( r neq 0 ), we have:[8 r h - h^2 - 16 r^2 = 0]Rearrange:[-h^2 + 8 r h - 16 r^2 = 0]Multiply both sides by -1:[h^2 - 8 r h + 16 r^2 = 0]This is a quadratic equation in ( h ):[h^2 - 8 r h + 16 r^2 = 0]Let me solve for ( h ):Using quadratic formula:[h = frac{8 r pm sqrt{(8 r)^2 - 4 cdot 1 cdot 16 r^2}}{2} = frac{8 r pm sqrt{64 r^2 - 64 r^2}}{2} = frac{8 r pm 0}{2} = 4 r]So, ( h = 4 r ). That's the height of the complete cone.Now, using the volume equation ( R^2 h = 8 r^3 ):[R^2 cdot 4 r = 8 r^3 implies R^2 = frac{8 r^3}{4 r} = 2 r^2 implies R = r sqrt{2}]So, the radius of the base of the complete cone is ( r sqrt{2} ).Alright, that takes care of the complete cone. Now, moving on to the truncated cone, which is a frustum.A frustum is a cone with the top cut off by a plane parallel to the base. The volume of a frustum is given by:[V = frac{1}{3} pi h (R^2 + R r + r^2)]Where ( R ) is the radius of the base, ( r ) is the radius of the top, and ( h ) is the height of the frustum.In this case, the volume of the frustum is also twice the volume of the sphere:[frac{1}{3} pi h (R^2 + R r + r^2) = frac{8}{3} pi r^3]Simplify:[h (R^2 + R r + r^2) = 8 r^3]Now, similar to the complete cone, the sphere is enclosed within the frustum. So, the sphere is tangent to both the top and bottom bases of the frustum and also tangent to the sides.Let me denote the radius of the base of the frustum as ( R ) and the radius of the top as ( r_t ). The height of the frustum is ( h_f ).Wait, but in the problem statement, it's mentioned that the sphere is enclosed within the frustum, so the sphere must fit perfectly inside, touching both the top, bottom, and the sides.This is a bit more complex. Let me think about how to relate the dimensions of the frustum to the sphere.First, the sphere touches the top and bottom bases of the frustum. So, the distance between the top and bottom bases (the height of the frustum) must be equal to the diameter of the sphere, which is ( 2 r ). So, ( h_f = 2 r ).Wait, is that correct? If the sphere is touching both the top and bottom, then the height of the frustum is equal to the diameter of the sphere. That seems logical.So, ( h_f = 2 r ).Now, the sphere is also tangent to the sides of the frustum. So, similar to the complete cone, we can use similar triangles to relate the radii of the top and bottom of the frustum to the sphere's radius.Let me denote ( R ) as the radius of the base and ( r_t ) as the radius of the top of the frustum.The frustum can be thought of as a portion of a larger cone. If we extend the frustum to form a complete cone, the height of the complete cone would be ( H = h_f + h ), where ( h ) is the height of the part that was cut off.But since the sphere is tangent to the sides of the frustum, the relationship between ( R ), ( r_t ), and ( h_f ) can be found using similar triangles.Let me consider the larger cone before truncation. The height of the larger cone is ( H = h + h_f ), and its base radius is ( R ). The frustum is created by cutting off a smaller cone of height ( h ) and base radius ( r_t ).Since the two cones (the larger one and the smaller one) are similar, the ratios of their corresponding dimensions are equal:[frac{r_t}{R} = frac{h}{H} = frac{h}{h + h_f}]Given that ( h_f = 2 r ), we have:[frac{r_t}{R} = frac{h}{h + 2 r}]Let me denote ( k = frac{r_t}{R} ). Then:[k = frac{h}{h + 2 r} implies h = frac{k (h + 2 r)}{1 - k}]Wait, that seems a bit convoluted. Maybe I should express ( h ) in terms of ( k ):From ( k = frac{h}{h + 2 r} ), cross-multiplying:[k (h + 2 r) = h implies k h + 2 k r = h implies h (1 - k) = 2 k r implies h = frac{2 k r}{1 - k}]So, ( h = frac{2 k r}{1 - k} ).Now, going back to the volume of the frustum:[h_f (R^2 + R r_t + r_t^2) = 8 r^3]We know ( h_f = 2 r ), so:[2 r (R^2 + R r_t + r_t^2) = 8 r^3 implies R^2 + R r_t + r_t^2 = 4 r^2]But ( r_t = k R ), so substituting:[R^2 + R (k R) + (k R)^2 = 4 r^2 implies R^2 (1 + k + k^2) = 4 r^2]So,[R^2 = frac{4 r^2}{1 + k + k^2}]Now, we also have the relationship from the similar triangles involving the sphere.The sphere is tangent to the sides of the frustum. So, similar to the complete cone, the distance from the center of the sphere to the side of the frustum is equal to the radius ( r ).But the frustum is part of a larger cone, so perhaps we can use similar reasoning as before.Let me consider the cross-sectional view of the frustum and the sphere. The frustum is a trapezoid in cross-section, with the sphere inscribed in it, touching both the top, bottom, and the sides.The center of the sphere is at the midpoint of the height of the frustum, which is ( r ) above the bottom and ( r ) below the top.In this cross-section, the sides of the frustum are two lines. The sphere is tangent to both lines.Let me denote the apex of the original cone as point ( A ), and the height of the original cone as ( H = h + h_f = h + 2 r ). The base radius is ( R ), and the radius at the height ( h ) (where the frustum starts) is ( r_t = k R ).The slope of the cone's side is ( frac{R}{H} ). At the height ( h ), the radius is ( r_t = k R ), so the slope is also ( frac{r_t}{h} = frac{k R}{h} ).Since the slopes must be equal:[frac{R}{H} = frac{k R}{h} implies frac{1}{H} = frac{k}{h} implies h = k H]But ( H = h + 2 r ), so:[h = k (h + 2 r) implies h = k h + 2 k r implies h (1 - k) = 2 k r implies h = frac{2 k r}{1 - k}]Which matches what I had earlier.Now, going back to the distance from the center of the sphere to the side of the frustum.In the cross-sectional view, the center of the sphere is at ( (0, r) ) if we consider the bottom of the frustum at ( y = 0 ) and the top at ( y = 2 r ). The sides of the frustum are lines connecting ( (R, 0) ) to ( (r_t, 2 r) ) and ( (-R, 0) ) to ( (-r_t, 2 r) ).Let me consider the right side of the frustum, which is the line connecting ( (R, 0) ) to ( (r_t, 2 r) ). The equation of this line can be found.The slope ( m ) is:[m = frac{2 r - 0}{r_t - R} = frac{2 r}{r_t - R}]So, the equation of the line is:[y - 0 = m (x - R) implies y = frac{2 r}{r_t - R} (x - R)]Simplify:[y = frac{2 r}{r_t - R} x - frac{2 r R}{r_t - R}]Now, the distance from the center ( (0, r) ) to this line should be equal to ( r ).Using the distance formula:[frac{|A x_0 + B y_0 + C|}{sqrt{A^2 + B^2}} = r]First, let me write the equation of the line in standard form ( A x + B y + C = 0 ).From above:[y = frac{2 r}{r_t - R} x - frac{2 r R}{r_t - R}]Multiply both sides by ( r_t - R ):[(r_t - R) y = 2 r x - 2 r R]Rearrange:[2 r x - (r_t - R) y - 2 r R = 0]So, ( A = 2 r ), ( B = -(r_t - R) ), ( C = -2 r R ).Now, the distance from ( (0, r) ) to this line is:[frac{|2 r cdot 0 - (r_t - R) cdot r - 2 r R|}{sqrt{(2 r)^2 + (-(r_t - R))^2}} = r]Simplify the numerator:[|0 - r (r_t - R) - 2 r R| = | - r r_t + r R - 2 r R | = | - r r_t - r R | = | - r (r_t + R) | = r (r_t + R)]Since distances are positive.The denominator:[sqrt{(2 r)^2 + (r_t - R)^2} = sqrt{4 r^2 + (r_t - R)^2}]So, the distance equation becomes:[frac{r (r_t + R)}{sqrt{4 r^2 + (r_t - R)^2}} = r]Divide both sides by ( r ):[frac{r_t + R}{sqrt{4 r^2 + (r_t - R)^2}} = 1]Multiply both sides by the denominator:[r_t + R = sqrt{4 r^2 + (r_t - R)^2}]Square both sides:[(r_t + R)^2 = 4 r^2 + (r_t - R)^2]Expand both sides:Left side:[r_t^2 + 2 r_t R + R^2]Right side:[4 r^2 + r_t^2 - 2 r_t R + R^2]Set them equal:[r_t^2 + 2 r_t R + R^2 = 4 r^2 + r_t^2 - 2 r_t R + R^2]Subtract ( r_t^2 + R^2 ) from both sides:[2 r_t R = 4 r^2 - 2 r_t R]Bring all terms to one side:[2 r_t R + 2 r_t R - 4 r^2 = 0 implies 4 r_t R - 4 r^2 = 0 implies 4 r_t R = 4 r^2 implies r_t R = r^2]So, ( r_t R = r^2 ). That's an important relationship.We also have from earlier:[R^2 + R r_t + r_t^2 = 4 r^2]But since ( r_t R = r^2 ), we can express ( r_t = frac{r^2}{R} ). Substitute this into the equation:[R^2 + R left( frac{r^2}{R} right ) + left( frac{r^2}{R} right )^2 = 4 r^2]Simplify term by term:First term: ( R^2 )Second term: ( R cdot frac{r^2}{R} = r^2 )Third term: ( frac{r^4}{R^2} )So, the equation becomes:[R^2 + r^2 + frac{r^4}{R^2} = 4 r^2]Subtract ( 4 r^2 ) from both sides:[R^2 + r^2 + frac{r^4}{R^2} - 4 r^2 = 0 implies R^2 - 3 r^2 + frac{r^4}{R^2} = 0]Multiply both sides by ( R^2 ) to eliminate the denominator:[R^4 - 3 r^2 R^2 + r^4 = 0]Let me set ( u = R^2 ), so the equation becomes:[u^2 - 3 r^2 u + r^4 = 0]This is a quadratic in ( u ):[u^2 - 3 r^2 u + r^4 = 0]Using the quadratic formula:[u = frac{3 r^2 pm sqrt{(3 r^2)^2 - 4 cdot 1 cdot r^4}}{2} = frac{3 r^2 pm sqrt{9 r^4 - 4 r^4}}{2} = frac{3 r^2 pm sqrt{5 r^4}}{2} = frac{3 r^2 pm r^2 sqrt{5}}{2}]So,[u = frac{r^2 (3 pm sqrt{5})}{2}]Since ( u = R^2 ), we have:[R^2 = frac{r^2 (3 + sqrt{5})}{2} quad text{or} quad R^2 = frac{r^2 (3 - sqrt{5})}{2}]Since ( R ) is a radius, it must be positive. Also, ( R > r_t ) because it's the base radius of the frustum. Let's check both solutions.First solution:[R^2 = frac{r^2 (3 + sqrt{5})}{2} implies R = r sqrt{frac{3 + sqrt{5}}{2}}]Second solution:[R^2 = frac{r^2 (3 - sqrt{5})}{2} implies R = r sqrt{frac{3 - sqrt{5}}{2}}]Now, let's compute ( r_t ) for each case using ( r_t = frac{r^2}{R} ).First case:[r_t = frac{r^2}{r sqrt{frac{3 + sqrt{5}}{2}}} = frac{r}{sqrt{frac{3 + sqrt{5}}{2}}} = r sqrt{frac{2}{3 + sqrt{5}}}]Rationalize the denominator:[sqrt{frac{2}{3 + sqrt{5}}} = sqrt{frac{2 (3 - sqrt{5})}{(3 + sqrt{5})(3 - sqrt{5})}} = sqrt{frac{2 (3 - sqrt{5})}{9 - 5}} = sqrt{frac{2 (3 - sqrt{5})}{4}} = sqrt{frac{3 - sqrt{5}}{2}}]So,[r_t = r sqrt{frac{3 - sqrt{5}}{2}}]Second case:[r_t = frac{r^2}{r sqrt{frac{3 - sqrt{5}}{2}}} = frac{r}{sqrt{frac{3 - sqrt{5}}{2}}} = r sqrt{frac{2}{3 - sqrt{5}}}]Rationalize the denominator:[sqrt{frac{2}{3 - sqrt{5}}} = sqrt{frac{2 (3 + sqrt{5})}{(3 - sqrt{5})(3 + sqrt{5})}} = sqrt{frac{2 (3 + sqrt{5})}{9 - 5}} = sqrt{frac{2 (3 + sqrt{5})}{4}} = sqrt{frac{3 + sqrt{5}}{2}}]So,[r_t = r sqrt{frac{3 + sqrt{5}}{2}}]Now, let's analyze both cases.Case 1:[R = r sqrt{frac{3 + sqrt{5}}{2}}, quad r_t = r sqrt{frac{3 - sqrt{5}}{2}}]Case 2:[R = r sqrt{frac{3 - sqrt{5}}{2}}, quad r_t = r sqrt{frac{3 + sqrt{5}}{2}}]But in the frustum, the base radius ( R ) must be larger than the top radius ( r_t ). Let's check the numerical values.Compute ( sqrt{frac{3 + sqrt{5}}{2}} ):First, ( sqrt{5} approx 2.236 ), so ( 3 + sqrt{5} approx 5.236 ). Then, ( frac{5.236}{2} approx 2.618 ). So, ( sqrt{2.618} approx 1.618 ).Similarly, ( sqrt{frac{3 - sqrt{5}}{2}} ):( 3 - sqrt{5} approx 0.764 ). Then, ( frac{0.764}{2} approx 0.382 ). So, ( sqrt{0.382} approx 0.618 ).So, in Case 1:[R approx 1.618 r, quad r_t approx 0.618 r]In Case 2:[R approx 0.618 r, quad r_t approx 1.618 r]But since ( R ) is the base radius and ( r_t ) is the top radius, and the frustum is wider at the base, Case 1 makes sense where ( R > r_t ). Case 2 would imply the top radius is larger than the base, which isn't typical for a frustum, unless it's an inverted frustum. But since the problem doesn't specify, I think both cases are mathematically valid, but physically, we usually consider ( R > r_t ). So, I'll proceed with Case 1.Therefore, the radii are:[R = r sqrt{frac{3 + sqrt{5}}{2}}, quad r_t = r sqrt{frac{3 - sqrt{5}}{2}}]Simplify these expressions:Note that ( sqrt{frac{3 + sqrt{5}}{2}} ) is actually the golden ratio ( phi = frac{1 + sqrt{5}}{2} approx 1.618 ). Let me verify:[phi = frac{1 + sqrt{5}}{2} approx 1.618]Compute ( sqrt{frac{3 + sqrt{5}}{2}} ):Let me square ( phi ):[phi^2 = left( frac{1 + sqrt{5}}{2} right )^2 = frac{1 + 2 sqrt{5} + 5}{4} = frac{6 + 2 sqrt{5}}{4} = frac{3 + sqrt{5}}{2}]So, ( sqrt{frac{3 + sqrt{5}}{2}} = phi ).Similarly, ( sqrt{frac{3 - sqrt{5}}{2}} ):Note that ( frac{3 - sqrt{5}}{2} = left( frac{sqrt{5} - 1}{2} right )^2 ).Compute ( left( frac{sqrt{5} - 1}{2} right )^2 ):[left( frac{sqrt{5} - 1}{2} right )^2 = frac{5 - 2 sqrt{5} + 1}{4} = frac{6 - 2 sqrt{5}}{4} = frac{3 - sqrt{5}}{2}]So, ( sqrt{frac{3 - sqrt{5}}{2}} = frac{sqrt{5} - 1}{2} ).Therefore, the radii can be written as:[R = r cdot phi = r cdot frac{1 + sqrt{5}}{2}, quad r_t = r cdot frac{sqrt{5} - 1}{2}]So, ( R = frac{r (1 + sqrt{5})}{2} ) and ( r_t = frac{r (sqrt{5} - 1)}{2} ).Alright, so now I have the dimensions of both the complete cone and the frustum.Now, the last part is to determine the ratio of the tangents of their slant heights.First, for the complete cone, the slant height ( l ) is given by:[l = sqrt{R^2 + h^2}]We have ( R = r sqrt{2} ) and ( h = 4 r ), so:[l = sqrt{(r sqrt{2})^2 + (4 r)^2} = sqrt{2 r^2 + 16 r^2} = sqrt{18 r^2} = 3 r sqrt{2}]The tangent of the angle ( alpha ) between the slant height and the base is:[tan alpha = frac{h}{R} = frac{4 r}{r sqrt{2}} = frac{4}{sqrt{2}} = 2 sqrt{2}]Wait, actually, the tangent of the angle between the slant height and the axis is ( frac{R}{h} ), but the problem says "tangents of their slant heights." Maybe I need to clarify.Wait, the slant height is a line, so the tangent of the angle it makes with the base is ( frac{h}{R} ), as I did above, which is ( 2 sqrt{2} ).For the frustum, the slant height ( l_f ) is given by:[l_f = sqrt{(R - r_t)^2 + h_f^2}]We have ( R = frac{r (1 + sqrt{5})}{2} ), ( r_t = frac{r (sqrt{5} - 1)}{2} ), and ( h_f = 2 r ).Compute ( R - r_t ):[R - r_t = frac{r (1 + sqrt{5})}{2} - frac{r (sqrt{5} - 1)}{2} = frac{r (1 + sqrt{5} - sqrt{5} + 1)}{2} = frac{2 r}{2} = r]So, ( R - r_t = r ).Therefore, the slant height:[l_f = sqrt{r^2 + (2 r)^2} = sqrt{r^2 + 4 r^2} = sqrt{5 r^2} = r sqrt{5}]The tangent of the angle ( beta ) between the slant height and the base is:[tan beta = frac{h_f}{R - r_t} = frac{2 r}{r} = 2]So, the tangent of the slant height for the complete cone is ( 2 sqrt{2} ), and for the frustum, it's ( 2 ).Therefore, the ratio of the tangents is:[frac{tan alpha}{tan beta} = frac{2 sqrt{2}}{2} = sqrt{2}]So, the ratio is ( sqrt{2} : 1 ).Let me recap:- Complete cone: height ( 4 r ), base radius ( r sqrt{2} ), tangent of slant height ( 2 sqrt{2} ).- Frustum: base radius ( frac{r (1 + sqrt{5})}{2} ), top radius ( frac{r (sqrt{5} - 1)}{2} ), tangent of slant height ( 2 ).- Ratio of tangents: ( sqrt{2} : 1 ).I think that covers all the required parts of the problem.

💡Okay, so I have this function f(x) defined in a piecewise manner. Let me try to understand it step by step. The function is given as:f(x) = { f(x + 1) if x < 1, 3^x if x ≥ 1}So, if x is less than 1, the function f(x) is equal to f(x + 1). That seems like a recursive definition. If x is 1 or greater, then f(x) is just 3 raised to the power of x.The question is asking for f(-1 + log₃5). Hmm, okay. Let me figure out what -1 + log base 3 of 5 is approximately. I know that log base 3 of 5 is the exponent to which I need to raise 3 to get 5. Since 3^1 = 3 and 3^2 = 9, log₃5 is somewhere between 1 and 2. Let me calculate it more precisely.Using the change of base formula, log₃5 = ln5 / ln3. Calculating that, ln5 is approximately 1.6094 and ln3 is approximately 1.0986. So, log₃5 ≈ 1.6094 / 1.0986 ≈ 1.46497. Therefore, -1 + log₃5 ≈ -1 + 1.46497 ≈ 0.46497.So, -1 + log₃5 is approximately 0.465, which is between 0 and 1. That means x = -1 + log₃5 is less than 1. Therefore, according to the piecewise function, f(x) = f(x + 1). So, f(-1 + log₃5) = f(0.465 + 1) = f(1.465).Now, 1.465 is greater than 1, so we use the other part of the piecewise function: f(x) = 3^x. Therefore, f(1.465) = 3^{1.465}.Wait, but 1.465 is approximately log₃5, right? Because earlier, we saw that log₃5 ≈ 1.46497. So, 1.465 is almost log₃5. So, 3^{1.465} is approximately 5. Because 3^{log₃5} = 5 by the definition of logarithms.Therefore, f(-1 + log₃5) = f(1.465) = 3^{1.465} ≈ 5.Let me verify that step again. So, starting with x = -1 + log₃5 ≈ 0.465, which is less than 1, so f(x) = f(x + 1). Then, x + 1 = 0.465 + 1 = 1.465, which is greater than or equal to 1, so f(1.465) = 3^{1.465}.But 1.465 is approximately log₃5, so 3^{log₃5} = 5. Therefore, f(-1 + log₃5) = 5.Looking at the options, C is 5, so that should be the correct answer.Wait, let me make sure I didn't make a mistake in the calculation. So, log₃5 is approximately 1.46497, so -1 + log₃5 is approximately 0.46497, which is indeed less than 1. So, we have to add 1 to it, getting approximately 1.46497, which is log₃5. Then, 3^{log₃5} is 5. So, yes, that seems correct.Alternatively, maybe I can think about this without approximating. Let's see. Let me denote y = -1 + log₃5. Then, y + 1 = log₃5, so f(y) = f(y + 1) = f(log₃5). Since log₃5 is greater than 1, f(log₃5) = 3^{log₃5} = 5. So, that's another way to see it without approximating.Therefore, the answer is 5, which is option C.**Final Answer**boxed{C}

💡First, I need to find the length of the garden. I know the perimeter of the garden is 64 meters and its width is 8 meters. Using the perimeter formula for a rectangle, Perimeter = 2 × (width + length), I can set up the equation:64 = 2 × (8 + length)Dividing both sides by 2 gives:32 = 8 + lengthSubtracting 8 from both sides:length = 24 metersNow that I have the length of the garden, I can calculate its area:Area of the garden = width × length = 8 × 24 = 192 square metersSince the playground has the same area, its area is also 192 square meters. The playground is 16 meters long, so I can find its width using the area formula:Area = length × width192 = 16 × widthDividing both sides by 16:width = 12 metersTherefore, the width of the playground is 12 meters.

💡Okay, so I'm trying to help my friends Artur and Timur decide which tax object to choose under the simplified taxation system (STS). They have two options: a) Income with a rate of 6%; b) Income minus expenses, with a rate of 15%, or a minimum tax of 1% of income.I need to figure out which option is better for them. Let's start by understanding what each option means.Option a) is straightforward: they pay 6% tax on their total income. That seems simple enough. But I wonder if there are any deductions or expenses they can subtract from their income before calculating the tax. The problem doesn't specify, so I'll assume they can't subtract any expenses under this option.Option b) is a bit more complicated. They can subtract their expenses from their income and then pay 15% tax on the remaining amount. However, there's also a minimum tax of 1% of their total income. So, they have to pay whichever is higher: 15% of (income minus expenses) or 1% of total income.To decide which option is better, I need to calculate the tax for both options and see which one results in a lower tax liability. First, I need to know their total income and expenses. The problem mentions "planned revenue" and "advances received," which sum up to 150,000,000 rubles. Their monthly expenses are listed as follows:- Rent: 770,000 rubles- Various oils: 1,450,000 rubles- Salaries: 4,600,000 rubles- Insurance contributions: 1,380,000 rubles- Accounting services: 340,000 rubles- Advertising: 1,800,000 rubles- Re-training: 800,000 rubles- Miscellaneous: 650,000 rublesLet me add these up to find their total monthly expenses:770,000 + 1,450,000 + 4,600,000 + 1,380,000 + 340,000 + 1,800,000 + 800,000 + 650,000 = 11,790,000 rubles per month.Since these are monthly expenses, I need to annualize them to compare with their annual income. So, multiplying by 12:11,790,000 * 12 = 141,480,000 rubles per year.Now, their total income is 150,000,000 rubles, and their total expenses are 141,480,000 rubles. For option a), the tax would be 6% of their total income:Tax_a = 150,000,000 * 0.06 = 9,000,000 rubles.For option b), the tax is 15% of (income minus expenses):Tax_b1 = (150,000,000 - 141,480,000) * 0.15 = 8,520,000 * 0.15 = 1,278,000 rubles.However, there's also a minimum tax of 1% of total income:Tax_b2 = 150,000,000 * 0.01 = 1,500,000 rubles.Since 1,500,000 is higher than 1,278,000, they would have to pay 1,500,000 rubles under option b).Comparing both options:- Option a): 9,000,000 rubles- Option b): 1,500,000 rublesClearly, option b) results in a significantly lower tax liability. But wait, I should double-check if there are any other factors to consider. For example, are there any limits on the expenses they can deduct under option b)? The problem doesn't specify, so I'll assume all listed expenses are deductible.Also, I need to ensure that the minimum tax is correctly calculated. It's 1% of total income, which is straightforward.Another thing to consider is the possibility of tax deductions or credits that might affect the final tax liability, but again, the problem doesn't mention any, so I'll proceed without them.In conclusion, based on the calculations, option b) is more advantageous for Artur and Timur as it results in a lower tax payment.

💡First, I need to determine the total number of students in the class by considering those taking French, German, both, and neither.I know that:- 41 students are taking French.- 22 students are taking German.- 9 students are taking both French and German.- 15 students are not taking either French or German.To avoid double-counting the students taking both courses, I'll use the principle of inclusion-exclusion. This means I'll add the number of students taking French to those taking German and then subtract those taking both.Calculating the total number of students taking at least one course:41 (French) + 22 (German) - 9 (Both) = 54 students.Finally, I'll add the students not taking either course to find the total number of students in the class:54 (at least one course) + 15 (neither course) = 69 students.

💡First, I need to determine the cost of one bottle of water. I'll start by identifying the given information:- The cost of one cola bottle is 3.- The cost of one juice bottle is 1.5.- The store sold 15 cola bottles, 25 water bottles, and 12 juice bottles.- The total revenue for the day was 88.Next, I'll calculate the total revenue from cola and juice sales:- Total revenue from cola: 15 bottles × 3 = 45.- Total revenue from juice: 12 bottles × 1.5 = 18.Adding these together gives the combined revenue from cola and juice: 45 + 18 = 63.To find the revenue from water sales, I'll subtract the combined revenue from cola and juice from the total revenue: 88 - 63 = 25.Since 25 bottles of water were sold, the cost of one bottle of water is: 25 ÷ 25 bottles = 1 per bottle.