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3 years ago

What is the surface area of a cone that has a slant height of 18.5 inches and a radius of 11 inches?

Mathematics

2 answers:

kykrilka [37]3 years ago

5 0

Answer: The answer is C. 324.5

Step-by-step explanation:

correct on edge

hammer [34]3 years ago

3 0

Where,

r is the radius

h is the height

l is the slant height

The area of the curved (lateral) surface of a cone = πrl

Note:

A cone does not have uniform (or congruent) cross-sections. (more about conic section here)

Example 1: A cone has a radius of 3cm and height of 5cm, find total surface area of the cone.

Solution:

To begin with we need to find slant height of the cone, which is determined by using Pythagoras, since the cross section is a right triangle.

l2 = h2 + r2

l2 = 52 + 32

l2 = 25 + 9

l = √(34)

l = 5.83 cm

And the total surface area of the cone is:

SA = πr2 + πrl

SA = π · r · (r + l)

SA = π · 3 · (3 + 5.83)

SA = 83.17 cm2

Therefore, the total surface area of the cone is 83.17cm2

Example 2: The total surface area of a cone is 375 square inches. If its slant height is four times the radius, then what is the base diameter of the cone? Use π = 3.

Solution:

The total surface area of a cone = πrl + πr2 = 375 inch2

Slant height: l = 4 × radius = 4r

Substitute l = 4r and π = 3

3 × r × 4 r + 3 × r2 = 375

12r2 + 3r2 = 375

15r2 = 375

r2 = 25

r = 25

r = 5

So the base radius of the cone is 5 inch.

And the base diameter of the cone = 2 × radius = 2 × 5 = 10 inch.

Example 3: What is the total surface area of a cone if its radius = 4cm and height = 3 cm.

Solution:

As mentioned earlier the formula for the surface area of a cone is given by:

SA = πr2 + πrl

SA = πr(r + l)

As in the previous example the slant can be determined using Pythagoras:

l2 = h2 + r2

l2 = 32 + 42

l2 = 9 + 16

l = 5

Insert l = 5 we will get:

SA = πr(r + l)

SA = 3.14 · 4 · (4+5)

SA = 113.04 cm2

Example 4: The slant height of a cone is 20cm. the diameter of the base is 15cm. Find the curved surface area of cone.

Solution:

Given that,

Slant height: l = 20cm

Diameter: d = 15cm

Radius: r = d/2 = 15/2 = 7.5cm

Curved surface area = πrl

CSA = πrl

CSA =π · 7.5 · 20

CSA =471.24 cm2

Example 5: Height and radius of the cone is 5 yard and 7 yard. Find the lateral surface area of the given cone.

Solution:

Lateral surface area of the cone = πrl

Step 1:

Slant height of the cone:

l2 = h2 + r2

l2 = 72 + 52

l2 = 49 + 25

l = 8.6

Step 2: Lateral surface area:

LSA = πrl

LSA = 3.14 × 7 × 8.6

LSA =189.03 yd2

So, the lateral surface area of the cone = 189.03 squared yard.

Example 6: A circular cone is 15 inches high and the radius of the base is 20 inches What is the lateral surface area of the cone?

Solution:

The lateral surface area of cone is given by:

LSA = π × r × l

LSA =3.14 × 20 × 15

LSA = 942 inch2

Example 7: Find the total surface area of a cone, whose base radius is 3 cm and the perpendicular height is 4 cm.

Solution:

Given that:

r = 3 cm

h = 4 cm

To find the total surface area of the cone, we need slant height of the cone, instead the perpendicular height.

The slant height l can be found by using Pythagoras theorem.

l2 = h2 + r2

l2 = 32 + 42

l2 = 9 + 16

l = 5

The total surface area of the cone is therefore:

SA = πr(r + l)

SA = 3.14 · 3 · (3+5)

SA = 75.36 cm2

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3 years ago

Read 2 more answers

Find the area of the following<br> kite:<br> A = [?] m²<br> 40 m<br> 16 m<br> 16 m<br> 6 m

Rama09 [41]

Answer:

$Area_{kite}=736m^2$

Step-by-step explanation:

There are a few methods to find the area of this figure:

1. kite area formula

2. 2 triangles (one top, one bottom)

3. 2 triangles (one left, one right)

4. 4 separate right triangles.

<h3><u>Option 1: The kite area formula</u></h3>

Recall the formula for area of a kite: $Area_{kite}=\frac{1}{2} d_{1}d_{2}$ where d1 and d2 are the lengths of the diagonals of the kite ("diagonals" are segments that connect non-adjacent vertices -- in a quadrilateral, vertices that are across from each other).

If you've forgotten why that is the formula for the area of a kite, observe the attached diagram: note that the kite (shaded in) is half of the area of the rectangle that surrounds the kite (visualize the 4 smaller rectangles, and observe that the shaded portion is half of each, and thus the area of the kite is half the area of the large rectangle).

The area of a rectangle is $Area_{rectangle}=bh$ , sometimes written as $Area_{rectangle}=bh$ , where w is the width, and h is the height of the rectangle.

In the diagram, notice that the width and height are each just the diagonals of the kite. So, the <u>Area of the kite</u> is <u>half of the area of that surrounding rectangle</u> ... the rectangle with sides the lengths of the kite's diagonals.Hence, $Area_{kite}=\frac{1}{2} d_{1}d_{2}$

For our situation, each of the diagonals is already broken up into two parts from the intersection of the diagonals. To find the full length of the diagonal, add each part together:

For the horizontal diagonal (which I'll call d1): $d_{1}=40m+6m=46m$

For the vertical diagonal (which I'll call d2): $d_{2}=16m+16m=32m$

Substituting back into the formula for the area of a kite:

$Area_{kite}=\frac{1}{2} d_{1}d_{2}\\Area_{kite}=\frac{1}{2} (46m)(32m)\\Area_{kite}=736m^2$

<h3><u /></h3><h3><u>Option 2: The sum of the parts (version 1)</u></h3>

If one doesn't remember the formula for the area of a kite, and can't remember how to build it, the given shape could be visualized as 2 separate triangles, the given shape could be visualized as 2 separate triangles (one on top; one on bottom).

Visualizing it in this way produces two congruent triangles. Since the upper and lower triangles are congruent, they have the same area, and thus the area of the kite is double the area of the upper triangle.

Recall the formula for area of a triangle: $Area_{triangle}=\frac{1}{2} bh$ where b is the base of a triangle, and h is the height of the triangle <em>(length of a perpendicular line segment between a point on the line containing the base, and the non-colinear vertex)</em>. Since all kites have diagonals that are perpendicular to each other (as already indicated in the diagram), the height is already given (16m).

The base of the upper triangle, is the sum of the two segments that compose it: $b=40m+6m=46m$

<u>Finding the Area of the upper triangle</u> $Area_{\text{upper }triangle}=\frac{1}{2} (46m)(16m) = 368m^2$

<u>Finding the Area of the kite</u>

$Area_{kite}=2*(368m^2)$

$Area_{kite}=736m^2$

<h3><u>Option 3: The sum of the parts (version 2)</u></h3>

The given shape could be visualized as 2 separate triangles (one on the left; one on the right). Each triangle has its own area, and the sum of both triangle areas is the area of the kite.

<em>Note: In this visualization, the two triangles are not congruent, so it is not possible to double one of their areas to find the area of the kite.</em>

The base of the left triangle is the vertical line segment the is the vertical diagonal of the kite. We'll need to add together the two segments that compose it: $b=16m+16m=32m$ . This is also the base of the triangle on the right.

<u>Finding the Area of left and right triangles</u>

$Area_{\text{left }triangle}=\frac{1}{2} (32m)(40m) = 640m^2$

The base of the right triangle is the same length as the left triangle: $Area_{\text{right }triangle}=\frac{1}{2} (32m)(6m) = 96m^2$

<u>Finding the Area of the kite</u>

$Area_{kite}=(640m^2)+(96m^2)$

$Area_{kite}=736m^2$

<h3><u>Option 4: The sum of the parts (version 3)</u></h3>

If you don't happen to see those composite triangles from option 2 or 3 when you're working this out on a particular problem, the given shape could be visualized as 4 separate right triangles, and we're still given enough information in this problem to solve it this way.

<u>Calculating the area of the 4 right triangles</u>

$Area_{\text{upper left }triangle}=\frac{1}{2} (40m)(16m) = 320m^2$