Which Rectangular Prism Has a Surface Area of 56 Square Units? note Art Is Not Drawn to Scale

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A rectangular prism has a book of lxx m³. If the length, width, and height of the prism are integers measured in meters, which of the following is NOT a possible measure of the area of the prism measured in square meters?

Caption:

Since the volume is the product of length, width, and height, and each of these three dimensions are integers, it is of import to know the factors of the volume. 70 = (2)(5)(7). This implies that each of these factors (and just these factors with the exception of 1) will evidence up in the iii dimensions exactly in one case. This creates precisely the post-obit five possibilities:

2, five, 7

SA = 2((ii)(v)+(2)(vii)+(5)(seven)) = 118

i, 7, ten

SA = two((1)(7)+(1)(10)+(seven)(x)) = 174

one, five, 14

SA = 2((1)(5)+(1)(14)+(5)(14)) = 178

1, ii, 35

SA = two((1)(2)+(1)(35)+(2)(35)) = 214

1, i, 70

SA = 2((1)(1)+(1)(70)+(1)(70)) = 282

The three sides of a rectangular box all take integer unit lengths. If each of the side lengths is greater than 1 unit of measurement, and if the volume of the box is 182 cubic units, what is the surface surface area of the box in foursquare units?

Caption:

Allow's phone call the side lengths of the box fifty, due west, and h. Nosotros are told that l, due west, and h must all be integer lengths greater than one. We are too told that the book of the box is 182 cubic units.

Since the volume of a rectangular box is the product of its side lengths, this ways that lwh must equal 182.

(l)(w)(h) = 182.

In order to determine possible values of l, w, and h, it would assist us to figure out the factors of 182. We want to express 182 as a product of three integers each greater than ane.

Let's cistron 182. Considering 182 is even, it is divisible past 2.

182 = ii(91).

91 is equal to the product of 7 and xiii.

Thus, 182 = 2(7)(13).

This means that the lengths of the box must be ii, 7, and 13 units.

In order to find the surface expanse, nosotros can utilize the following formula:

surface surface area = 2lw + 2lh + 2hw.

surface area = ii(ii)(7) + 2(ii)(xiii) + 2(7)(xiii)

= 28 + 52 + 182

= 262 square units.

The reply is 262.

A right rectangular prism has dimensions of iii x five 10 xx. What is its surface surface area?

Explanation:

In that location are six faces to a right, rectangular prism. Based on our dimensions, nosotros know that we must accept a face that is iii x 5, a face that is 5 x 20 and a face that is 3 x 20. To call up this through, imagine that the front end confront is 3 x 5, the right side is 5 x 20, and the superlative is 3 x 20. Now, each of these sides has a matching side opposite (the left has the right, the meridian has the bottom, the front has the back).

Therefore, we know we accept the following areas for the faces of our prism:

2 * 3 * 5 = xxx

ii * v * xx = 200

2 * 3 * 20 = 120

Add these to get the total expanse:

thirty + 200 + 120 = 350

A right rectangular prism has dimensions of 12.four x 2.3 x 33. What is its surface expanse?

Possible Answers:

1882.32

1027.24

513.62

941.16

470.58

Explanation:

There are vi faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 12.4 10 ii.three, a face that is 2.3 ten 33 and a face up that is 33 10 12.four. To think this through, imagine that the front face is 12.4 ten 2.3, the left side is 2.3 x 33, and the top is 33 x 12.4. Now, each of these sides has a matching side opposite (the left has the right, the height has the lesser, the front has the back).

Therefore, we know we have the following areas for the faces of our prism:

2 * 12.iv * 2.3 = 57.04

ii * ii.iii * 33 = 151.8

2 * 12.4 * 33 = 818.4

Add these to get the total surface area:

57.04 + 151.8 + 818.4 = 1027.24

The dimensions of a correct rectangular prism are such that the second dimension is twice the length of the start and the third is twice the length of the second. If the book of the prism is 216 cubic units, what is its surface area?

Possible Answers:

126 square units

None of the other answers

252 square units

189 square units

215 square units

Right answer:

252 foursquare units

Explanation:

Based on our prompt, we tin can say that the prism has dimensions that tin be represented equally:

Dim1: x

Dim2: 2 * Dim1 = 2x

Dim3: 2 * Dim2 = 2 * 2x = 4x

More direct stated, therefore, our dimensions are: x, 2x, and 4x. Therefore, the book is x * 2x * 4x = 216, which simplifies to 8x³ = 216 or x³ = 27. Solving for ten, we find 10 = 3. Therefore, our dimensions are:

10 = iii

2x = 6

4x = 12

Or: 3 x half dozen x 12

Now, to find the surface expanse, nosotros must consider that this ways that our prism has sides of the post-obit dimensions: 3 10 6, 6 x 12, and 3 x 12. Since each side has a "matching" side opposite it, nosotros know that we accept the following values for the areas of the faces:

ii * 3 * six = 36

2 * 6 * 12 = 144

2 * three * 12 = 72

The total surface surface area therefore equals: 36 + 144 + 72 = 252 square units.

The area of a given object is 30,096 in². What is the expanse of this object in ft²?

Possible Answers:

None of the other answers

1254 ft²

1881 ft²

2508 ft^two

209 ftⁱⁱ

Explanation:

Converting squared units is not difficult, though you have to be careful not to brand a simple mistake. It is tempting to think you lot tin simply divide the initial value (xxx,096) by 12, every bit though you were converting from inches to feet.

Begin by thinking this through every bit follows. In the example of a single dimension, we know that:

1 ft = 12 in or one in = (1/12) ft

At present, think the case of a foursquare with dimensions 1 ft x ane ft. This foursquare has the following dimensions in inches: 12 in x 12 in. The surface area is therefore 12 * 12 = 144 in². This holds for all ii-dimensional conversions. Therefore, the two dimensional conversion equation is:

1 ft² = 144 inⁱⁱ or 1 in² = (1/144) ft²

Based on this, we can convert our value thirty,096 in² thus: 30,096/144 = 209 ft².

The area of a given object is 24 ydⁱⁱ. What is the area of this object in in²?

Possible Answers:

31,104 in²

10,368 in²

None of the other answers

864 in²

xx,736 in²

Correct answer:

31,104 in²

Explanation:

Converting squared units is not difficult, though y'all take to be conscientious not to brand a uncomplicated mistake. It is tempting to call up y'all can merely multiply the initial value (24) past 36, as though you were converting from yards to inches.

Begin by thinking this through as follows. In the case of a single dimension, we know that:

1 yd = 36 in

Now, remember the case of a square with dimensions 1 yd x ane yd. This square has the following dimensions in inches: 36 in 10 36 in. The area is therefore 36 * 36 = 1296 in². This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:

1 yd² = 1296 inⁱⁱ

Based on this, we can convert our value 24 yd^two thus: 24 * 1296 = 31,104 in².

Angie is painting a two pes cube for a play she is in. She needs $25\hspace{1 mm}mL$ of pigment for every square pes she paints. How much paint does she need?

Possible Answers:

It is impossible to convert between metric units and feet.

$1.041\overline{6}\hspace{1 mm}mL$

$100\hspace{1 mm}mL$

None of the available answers

$600\hspace{1 mm}mL$

Correct respond:

$600\hspace{1 mm}mL$

Caption:

First we must calculate the surface expanse of the cube. Nosotros know that there are 6 surfaces and each surface has the same area:

$Area=6(2^2)=6\times 4=24\hspace{1 mm}feet^2$

Now we volition determine the corporeality of pigment needed

$24\hspace{1 mm}feet^2\times \frac{25\hspace{1 mm}mL}{1\hspace{1 mm}foot^2}=600\hspace{1 mm}mL$

What is the surface expanse of an equilateral triangluar prism with edges of 6 in and a height of 12 in?

Allow $\sqrt{3}=1.732$ and $\sqrt{2}=1.414$ .

Correct answer:

$247.176\ in^{2}$

Explanation:

The surface area of the prism tin exist broken into iii rectangular sides and two equilateral triangular bases.

The area of the sides is given by: $A=lw=12\cdot 6=72\ in^{2}$ , and then for all 3 sides nosotros become $\dpi{100} 216\ in^{2}$ .

The equilateral triangle is also an equiangular triangle by definition, so the base has congruent sides of half dozen in and three angles of 60 degrees. Nosotros use a special right traingle to figure out the height of the triangle: thirty - 60 - xc. The tiptop is the side opposite the 60 degree angle, and so information technology becomes $3\sqrt{3}$ or 5.196.

The expanse for a triangle is given by $A=\frac{1}{2}bh$ and since we need two of them we get $A=bh=6\cdot 3\sqrt{3}=31.176\ in^{2}$ .

Therefore the full surface area is $216\ in^{2}+31.176\ in^{2}=247.176\ in^{2}$ .

What is the total surface surface area of a boxwithout a lid, if the dimensions of the base of the box are $6\textup{ in by }10\textup{ in}$ , and the box is $5\textup{ in}$ tall?

Right answer:

$220\textup{ sq in}$

Explanation:

The area of a rectangular prism is 2lw+2lh+2wh . Our prism, however, is missing its top, then it will be. This gives us (10)(6)+2(10)(5)+2(6)(5)=220 .

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Which Rectangular Prism Has a Surface Area of 56 Square Units? note Art Is Not Drawn to Scale

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