The lateral area of a cone is $559 pi the radius is 16.8 CM what is the slant height to the nearest tenth of a centimeter
A 8.4 cm
B 12.9 cm
C 27.7 cm
D 33.3 cm
The lateral area of a cone is given by $L=\pi rs$, where $r$ is the radius and $s$ is the slant height. In this problem, we are given that $L=559\pi$ and $r=16.8$. We want to find $s$.
Substituting the given values, we have $559\pi = \pi (16.8)s$. Dividing both sides by $\pi$ gives $559=16.8s$. Solving for $s$, we get $s=\frac{559}{16.8}\approx 33.2$.
Rounding to the nearest tenth of a centimeter, we get $\boxed{\textbf{(D) }33.3\text{ cm}}$.
To find the slant height of a cone, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides. In the case of a cone, the slant height forms a right triangle with the height and radius of the cone.
Given:
Lateral area of the cone = 559π
Radius = 16.8 cm
To find the slant height, we need to find the height of the cone first.
The formula for the lateral area of a cone is given by:
Lateral area = πrℓ, where r is the radius and ℓ is the slant height.
From the given information, we can set up the equation as follows:
559π = π(16.8)(ℓ)
Simplifying the equation:
559 = 16.8ℓ
To find the height (not the slant height), we will use the formula for the surface area of a cone:
Surface area = πr(r + ℓ), where ℓ is the slant height.
From the given information, we can set up the equation as follows:
559π = π(16.8)(16.8 + ℓ)
Simplifying the equation:
559 = 16.8(16.8 + ℓ)
559 = 282.24 + 16.8ℓ
16.8ℓ = 559 - 282.24
16.8ℓ = 276.76
ℓ ≈ 16.45 cm
Now that we have the slant height, we need to round it to the nearest tenth of a centimeter.
The options given are:
A) 8.4 cm
B) 12.9 cm
C) 27.7 cm
D) 33.3 cm
Rounding 16.45 cm to the nearest tenth gives us the answer:
B) 12.9 cm
To find the slant height of a cone, we can use the Pythagorean Theorem. The slant height (l) of a cone, the height (h), and the radius (r) form a right triangle.
In this case, we are given the lateral area (L) of the cone, which is given by the formula L = πrl, where r is the radius of the cone's base and l is the slant height.
We are also given the radius (r) of the cone, which is 16.8 cm, and the lateral area (L), which is 559π cm².
To calculate the slant height (l), we can rearrange the formula for the lateral area:
L = πrl
559π = π(16.8)l
Divide both sides by π:
559 = 16.8l
Now, isolate l:
l = 559 / 16.8
l ≈ 33.244
Rounded to the nearest tenth, the slant height is approximately 33.3 cm.
Therefore, the correct answer is option D: 33.3 cm.