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Sagot :
To solve the given problem, we need to find the radius of the base, the height, and the volume of a cone with a slant height of 13 cm and a total surface area of 90π cm².
### Step 1: Find the Radius of the Base
The total surface area [tex]\(A\)[/tex] of a cone is given by the formula:
[tex]\[ A = \pi r (l + r) \][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(l\)[/tex] is the slant height.
We are given:
[tex]\[ A = 90\pi \][/tex]
[tex]\[ l = 13 \][/tex]
Substitute the known values into the formula:
[tex]\[ 90\pi = \pi r (13 + r) \][/tex]
Divide both sides by [tex]\(\pi\)[/tex]:
[tex]\[ 90 = r (13 + r) \][/tex]
Rearrange into a quadratic equation:
[tex]\[ r^2 + 13r - 90 = 0 \][/tex]
To solve the quadratic equation [tex]\(ar^2 + br + c = 0\)[/tex], we use the quadratic formula:
[tex]\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our equation, [tex]\(a = 1\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = -90\)[/tex].
Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 13^2 - 4(1)(-90) = 169 + 360 = 529 \][/tex]
Take the square root of the discriminant:
[tex]\[ \sqrt{529} = 23 \][/tex]
Now, apply the quadratic formula:
[tex]\[ r = \frac{-13 \pm 23}{2} \][/tex]
This yields two solutions:
[tex]\[ r_1 = \frac{-13 + 23}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ r_2 = \frac{-13 - 23}{2} = \frac{-36}{2} = -18 \][/tex]
Since the radius cannot be negative, we have:
[tex]\[ r = 5 \, \text{cm} \][/tex]
So, the radius of the base is [tex]\(5 \, \text{cm}\)[/tex].
### Step 2: Identify the Height of the Cone
To find the height [tex]\(h\)[/tex] of the cone, we use the Pythagorean theorem. For the cone:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
where [tex]\(l\)[/tex] is the slant height, [tex]\(r\)[/tex] is the radius, and [tex]\(h\)[/tex] is the height.
We have:
[tex]\[ l = 13 \, \text{cm} \][/tex]
[tex]\[ r = 5 \, \text{cm} \][/tex]
Substitute these values into the equation:
[tex]\[ 13^2 = 5^2 + h^2 \][/tex]
[tex]\[ 169 = 25 + h^2 \][/tex]
Solve for [tex]\(h^2\)[/tex]:
[tex]\[ h^2 = 169 - 25 \][/tex]
[tex]\[ h^2 = 144 \][/tex]
Take the square root of both sides:
[tex]\[ h = \sqrt{144} \][/tex]
[tex]\[ h = 12 \, \text{cm} \][/tex]
So, the height of the cone is [tex]\(12 \, \text{cm}\)[/tex].
### Step 3: Determine the Volume of the Cone
The volume [tex]\(V\)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
We already know:
[tex]\[ r = 5 \, \text{cm} \][/tex]
[tex]\[ h = 12 \, \text{cm} \][/tex]
Substitute these values into the formula:
[tex]\[ V = \frac{1}{3} \pi (5)^2 (12) \][/tex]
Calculate the volume:
[tex]\[ V = \frac{1}{3} \pi (25) (12) \][/tex]
[tex]\[ V = \frac{1}{3} \pi (300) \][/tex]
[tex]\[ V = 100\pi \][/tex]
So, the volume of the cone is [tex]\(100\pi \, \text{cm}^3\)[/tex].
In approximate numerical form:
[tex]\[ V \approx 100 \times 3.141592653589793 \][/tex]
[tex]\[ V \approx 314.16 \, \text{cm}^3 \][/tex]
### Summary
(i) The radius of the base of the cone is [tex]\(5 \, \text{cm}\)[/tex].
(ii) The height of the cone is [tex]\(12 \, \text{cm}\)[/tex].
(iii) The volume of the cone is approximately [tex]\(314.16 \, \text{cm}^3\)[/tex].
### Step 1: Find the Radius of the Base
The total surface area [tex]\(A\)[/tex] of a cone is given by the formula:
[tex]\[ A = \pi r (l + r) \][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(l\)[/tex] is the slant height.
We are given:
[tex]\[ A = 90\pi \][/tex]
[tex]\[ l = 13 \][/tex]
Substitute the known values into the formula:
[tex]\[ 90\pi = \pi r (13 + r) \][/tex]
Divide both sides by [tex]\(\pi\)[/tex]:
[tex]\[ 90 = r (13 + r) \][/tex]
Rearrange into a quadratic equation:
[tex]\[ r^2 + 13r - 90 = 0 \][/tex]
To solve the quadratic equation [tex]\(ar^2 + br + c = 0\)[/tex], we use the quadratic formula:
[tex]\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our equation, [tex]\(a = 1\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = -90\)[/tex].
Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 13^2 - 4(1)(-90) = 169 + 360 = 529 \][/tex]
Take the square root of the discriminant:
[tex]\[ \sqrt{529} = 23 \][/tex]
Now, apply the quadratic formula:
[tex]\[ r = \frac{-13 \pm 23}{2} \][/tex]
This yields two solutions:
[tex]\[ r_1 = \frac{-13 + 23}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ r_2 = \frac{-13 - 23}{2} = \frac{-36}{2} = -18 \][/tex]
Since the radius cannot be negative, we have:
[tex]\[ r = 5 \, \text{cm} \][/tex]
So, the radius of the base is [tex]\(5 \, \text{cm}\)[/tex].
### Step 2: Identify the Height of the Cone
To find the height [tex]\(h\)[/tex] of the cone, we use the Pythagorean theorem. For the cone:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
where [tex]\(l\)[/tex] is the slant height, [tex]\(r\)[/tex] is the radius, and [tex]\(h\)[/tex] is the height.
We have:
[tex]\[ l = 13 \, \text{cm} \][/tex]
[tex]\[ r = 5 \, \text{cm} \][/tex]
Substitute these values into the equation:
[tex]\[ 13^2 = 5^2 + h^2 \][/tex]
[tex]\[ 169 = 25 + h^2 \][/tex]
Solve for [tex]\(h^2\)[/tex]:
[tex]\[ h^2 = 169 - 25 \][/tex]
[tex]\[ h^2 = 144 \][/tex]
Take the square root of both sides:
[tex]\[ h = \sqrt{144} \][/tex]
[tex]\[ h = 12 \, \text{cm} \][/tex]
So, the height of the cone is [tex]\(12 \, \text{cm}\)[/tex].
### Step 3: Determine the Volume of the Cone
The volume [tex]\(V\)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
We already know:
[tex]\[ r = 5 \, \text{cm} \][/tex]
[tex]\[ h = 12 \, \text{cm} \][/tex]
Substitute these values into the formula:
[tex]\[ V = \frac{1}{3} \pi (5)^2 (12) \][/tex]
Calculate the volume:
[tex]\[ V = \frac{1}{3} \pi (25) (12) \][/tex]
[tex]\[ V = \frac{1}{3} \pi (300) \][/tex]
[tex]\[ V = 100\pi \][/tex]
So, the volume of the cone is [tex]\(100\pi \, \text{cm}^3\)[/tex].
In approximate numerical form:
[tex]\[ V \approx 100 \times 3.141592653589793 \][/tex]
[tex]\[ V \approx 314.16 \, \text{cm}^3 \][/tex]
### Summary
(i) The radius of the base of the cone is [tex]\(5 \, \text{cm}\)[/tex].
(ii) The height of the cone is [tex]\(12 \, \text{cm}\)[/tex].
(iii) The volume of the cone is approximately [tex]\(314.16 \, \text{cm}^3\)[/tex].
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