Solid Geometry - Geometry

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Question

Find the surface area of a cone with a base diameter of and a height of .

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Answer

The Surface Area of a cone is:

$SA=\pi rs + \pi r^2$

Given the base diameter is 6, the radius of the base is 3. The height is 10. We will substitute these values to find the slant height by using the Pythagorean Theorem.

$\sqrt{109} = s$

Substitute slant height and radius into the Surface Area equation.

$SA=\pi (3)(\sqrt{109}) + \pi (3)^2 = 3\pi \sqrt{109} + 9\pi$

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Question

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

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Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

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Question

The height of a cone is ; the diameter of its base is twice the height. Give its surface area in terms of .

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Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is twice the height, which is ; the radius is half this, which is .

The slant height can be calculated using the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h^{2}}$

$l = \sqrt{h^{2}+ h^{2}}$

$l = \sqrt{2h^{2}}$

$l =h \sqrt{2}$

Substitute $h \sqrt{2}$ for and for in the surface area formula:

$A = \pi h^{2} + \pi h \cdot h \sqrt{2}$

$A = \pi h^{2} + \pi h ^{2} \sqrt{2}$

$A = \left (1 + \sqrt{2} \right ) \pi h^{2}$

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Question

The radius of the base of a cone is ; its height is twice of the diameter of that base. Give its surface area in terms of .

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Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The base has radius and diameter . The height is twice the diamter, which is . Its slant height can be calculated using the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h^{2}}$

$l = \sqrt{r^{2}+ (4r)^{2}}$

$l = \sqrt{r^{2}+ 16r^{2}}$

$l = \sqrt{17r^{2}}$

$l = r \sqrt{17 }$

Substitute $r \sqrt{17 }$ for in the surface area formula:

$A = \pi r^{2} + \pi r \cdot r \sqrt{17}$

$A = \pi r^{2} + \pi r^{2} \sqrt{17}$

$A = \left ( 1+ \sqrt{17} \right ) \pi r^{2}$

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Question

The circumference of the base of a cone is 100; the height of the cone is equal to the diameter of the base. Give the surface area of the cone (nearest whole number).

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Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is the circumference divided by $\pi$ , which is

$d = C \div (2\pi ) \approx 100 \div 3.14 \approx 31.83$

This is also the height .

The radius is half this, or

$r = d \div 2 \approx 31.83 \div 2 \approx 15.92$

The slant height can be found by way of the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h ^{2}}$

$l \approx \sqrt{15.92^{2}+ 31.83 ^{2}}$

$l \approx \sqrt{253.30+ 1,013.21}$

$l \approx \sqrt{1266.51}$

$l \approx 35.60$

Substitute in the surface area formula:

$A = \pi r^{2} + \pi r l$

$A \approx 3.14 \cdot 15.92 ^{2} + 3.14 \cdot 15.92 \cdot 35.60$

$A \approx 795.8+ 1,780.1$

$A \approx 2,575$

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Question

The circumference of the base of a cone is 80; the slant height of the cone is equal to twice the diameter of the base. Give the surface area of the cone (nearest whole number).

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Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The slant height is twice the diameter, or, equivalently, four times the radius, so

and

$A = \pi r^{2} + \pi r \cdot 4r$

$A = \pi r^{2} +4 \pi r^{2}$

$A = 5 \pi r^{2}$

The radius of the base is the circumference divided by $2\pi$ , which is

$80 \div 2 \pi \approx 80 \div (2 \times 3.14 )\approx 12.73$

Substitute:

$A = 5 \pi r^{2}$

$A \approx 5 \cdot 3.14 \cdot (12.73)^{2}$

$A \approx 2,546$

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Question

The radius of the base of a cone is ; its slant height is two-thirds of the diameter of that base. Give its surface area in terms of .

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Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is twice radius , or , and its slant height is two-thirds of this diameter, which is $\frac{2}{3}\cdot 2r = \frac{4}{3}r$ . Substitute this for in the formula:

$A = \pi r^{2} + \pi r \cdot \frac{4}{3}r$

$A = \pi r^{2} + \frac{4}{3} \pi r ^{2}$

$A = \frac{7}{3} \pi r ^{2} = \frac{7\pi r ^{2}}{3}$

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Question

If a cone were unfurled into a 2-dimensional figure. The lateral area of the cone would look most like which figure?

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Answer

When creating a net image of a 3D figure - one imagines it is made of paper and is unfurled into its' 2D form. The lateral portion of the cone cone would be unfurled into the image of a Sector of a Circle. To include the full surface area of the cone a circle is included to form the base of the cone as in the figure below. The lateral area portion is the top part of the figure below.

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Question

As shown by the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(15)(14)\pi=210\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(14)(10)+\pi(14)^2=476\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=210\pi+476\pi=686\pi=2155.13$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(20)(12)\pi=240\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(12)(8)+\pi(12)^2=336\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=240\pi+336\pi=576\pi=1809.56$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(8)(4)\pi=32\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(4)(2)+\pi(4)^2=32\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=32\pi+32\pi=64\pi=201.06$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(13)(5)\pi=65\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(5)(9)+\pi(5)^2=115\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=65\pi+115\pi=180\pi=565.49$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(15)(4)\pi=60\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(4)(6)+\pi(4)^2=64\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=60\pi+64\pi=124\pi=389.56$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(9)(4)\pi=36\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(9)(4)+\pi(4)^2=88\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=36\pi+88\pi=124\pi=389.56$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(15)(10)\pi=150\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(10)(8)+\pi(10)^2=260\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=150\pi+260\pi=410\pi=1288.05$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(16)(12)\pi=192\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(12)(13)+\pi(12)^2=456\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=192\pi+456\pi=648\pi=2035.75$

Make sure to round to places after the decimal.

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Question

As shown by the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(13)(10)\pi=130\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(10)(12)+\pi(10)^2=340\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=130\pi+340\pi=470\pi=1476.55$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(25\sqrt3)(22)\pi=(550\sqrt3)\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(22)(20)+\pi(22)^2=1364\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=(550\sqrt3)\pi+1364\pi=7277.90$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(17)(10)\pi=170\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(10)(15)+(10)^2\pi=400\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=170\pi+400\pi=570\pi=1790.71$

Make sure to round to places after the decimal.

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Question

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

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Answer

First, find the lateral surface area of the cone.

$\text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})$

Plug in the given slant height and radius.

$\text{Lateral Area}=(18)(12)\pi=216\pi$

Next, find the surface area of the cylinder:

$\text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2$

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

$\text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2$

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

$\text{Surface Area of Cylindrical Portion}=2\pi(12)(15)+\pi(12)^2=504\pi$

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

$\text{Surface Area of Figure}=216\pi+504\pi=720\pi=2261.95$

Make sure to round to places after the decimal.

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