Question

Радиус меньшего основания усеченно конуса равен 6см,а образующая равно 5см, наклонена к плоскости основания под углом 60 градусов.Найти Sбок, Sполн.

Answer 1

R=O1A1=6см
L=A1A=5см
<A1A0=60
A1H_|_AO⇒ΔA1AO-прямоугольный⇒<AA1O=90-60=30⇒AH=1/2A1A=2,5см
R=OA=6+2,5=8,5см
Sбок=πL*(R+r)=π*5*(6+8,5)=5π*14,5=72,5πсм²
Sпол=Sбок+Sосн+S1осн=72,5π+36π+72,25π=180,75πсм²

Answer 2

Given Information

We are given the following information about a truncated cone: - The radius of the smaller base is 6 cm. - The slant height (or generatrix) is 5 cm. - The generatrix is inclined to the plane of the base at an angle of 60 degrees.

Solution

To find the lateral surface area (Sбок) and the total surface area (Sполн) of the truncated cone, we can use the following formulas:

1. Lateral Surface Area (Sбок): - The lateral surface area of a truncated cone can be calculated using the formula: Sбок = π(R1 + R2)l, where R1 and R2 are the radii of the larger and smaller bases, and l is the slant height. - In this case, the radius of the smaller base (R2) is given as 6 cm, and the slant height (l) is given as 5 cm. - We need to find the radius of the larger base (R1) to calculate the lateral surface area.

2. Total Surface Area (Sполн): - The total surface area of a truncated cone can be calculated using the formula: Sполн = Sбок + π(R1^2 + R2^2), where R1 and R2 are the radii of the larger and smaller bases. - We already have the value of Sбок from the previous step. - We need to find the radius of the larger base (R1) to calculate the total surface area.

Let's calculate the values step by step.

Calculating the Radius of the Larger Base (R1)

To find the radius of the larger base (R1), we can use the given information and trigonometric relationships.

From the given information, we know that the generatrix is inclined to the plane of the base at an angle of 60 degrees. Let's denote this angle as α.

Using trigonometry, we can relate the slant height (l), the radius of the smaller base (R2), and the radius of the larger base (R1) as follows:

cos(α) = (R1 - R2) / l

Substituting the given values, we have:

cos(60) = (R1 - 6) / 5

Simplifying the equation, we get:

(R1 - 6) / 5 = 1/2

Solving for R1, we have:

R1 - 6 = 5/2

R1 = 5/2 + 6

R1 = 5/2 + 12/2

R1 = 17/2

R1 = 8.5 cm

Now that we have the value of R1, we can proceed to calculate the lateral surface area (Sбок) and the total surface area (Sполн).

Calculating the Lateral Surface Area (Sбок)

Using the formula for the lateral surface area of a truncated cone, we have:

Sбок = π(R1 + R2)l

Substituting the given values, we get:

Sбок = π(8.5 + 6)(5)

Sбок = π(14.5)(5)

Sбок ≈ 72.25π cm^2

Calculating the Total Surface Area (Sполн)

Using the formula for the total surface area of a truncated cone, we have:

Sполн = Sбок + π(R1^2 + R2^2)

Substituting the given values, we get:

Sполн = 72.25π + π(8.5^2 + 6^2)

Sполн = 72.25π + π(72.25 + 36)

Sполн = 72.25π + π(108.25)

Sполн ≈ 180.5π cm^2

Answer

Therefore, the lateral surface area (Sбок) of the truncated cone is approximately 72.25π cm^2, and the total surface area (Sполн) is approximately 180.5π cm^2.

Please note that the values are approximate and have been calculated using the given information and mathematical formulas.