# A #5 cm# high prism has a right triangle base with cateti #6 cm and 8 cm# long. Calculate the surface area using a perimeter construct and a single formula. Calculate also the volume?

The term is actualli latin "catetus". the plural "cateti" us also used in italian as is. As an aside most of the Italian was derived from latin hence the close similarity... In general an ending like "us" is almost never italian. Italian nouns for the most part end with vowels - singular a or o and plural e or i depending on gender... Cheers

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Surface Area:

Volume:

Here is the prism as I believe it has been described:

The hypotenuse length of

Since we were asked to use a perimeter construct in calculating the surface area, we might note that the vertical surface area is:

The total surface area is the vertical surface area plus the area of the top and the bottom right triangles:

The volume of a prism is the area of its base times its height:

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As a side note:

The term *cateti* (singular *catetus*, if I remember correctly) it a term that refers to the non-hypotenuse sides of a right triangle. It is not a term we often see (at least in English, I'm not sure that I have encountered it in over fifty years) but I think it is a useful one and would encourage it as more convenient that saying "the non-hypotenuse sides". (I think the term may be Italian, if not Latin; could someone verify or correct this?)

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The surface area of the prism is 148 cm² and the volume is 160 cm³.

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To calculate the surface area of the prism, use the formula:

[ A = 2B + Ph ]

Where: ( B ) = area of the base ( P ) = perimeter of the base ( h ) = height of the prism

Given that the base is a right triangle with legs ( 6 ) cm and ( 8 ) cm, the area ( B ) of the base can be found using the formula for the area of a triangle:

[ B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 8 = 24 , \text{cm}^2 ]

The perimeter ( P ) of the base is the sum of the lengths of its three sides:

[ P = 6 + 8 + \sqrt{6^2 + 8^2} = 6 + 8 + 10 = 24 , \text{cm} ]

The height ( h ) of the prism is given as ( 5 ) cm.

Substitute these values into the formula for the surface area:

[ A = 2B + Ph = 2(24) + 24(5) = 48 + 120 = 168 , \text{cm}^2 ]

To find the volume of the prism, use the formula:

[ V = Bh ]

Substitute the area of the base ( B ) and the height ( h ) into the formula:

[ V = 24 \times 5 = 120 , \text{cm}^3 ]

So, the surface area of the prism is ( 168 , \text{cm}^2 ) and the volume is ( 120 , \text{cm}^3 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A triangle has two corners with angles of # pi / 12 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #8 #, what is the largest possible area of the triangle?
- A right trapezoidal prism has base dimensions of #40 cm and 56 cm# and a height of #30 cm#. Calculate the volume, surface area and the weight of the prism, knowing it is #120 cm# high and has a density of #2.5 gm/cm^3#?
- Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 3 #. If one side of the triangle has a length of # 12 #, what is the longest possible perimeter of the triangle?
- If area of a kite formed by #y=f(x)=-1/2|x+6|-4# and #y=g(x)=k|x+6|-10# is #18#, find #k#?
- What is the perimeter of the rectangle with sides 5#1/3# in and 10#1/6#in?

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