(3) Lines that forms straight angles

What is all of the surface area and volume of this Castle? Find the surface area and volume of all the figures below, then out o

motikmotik

Answer:

Step-by-step explanation:

There are a few formulas that are useful for this:

lateral area of a pyramid or cone: LA = 1/2·Ph, where P is the perimeter and h is the slant height
lateral area of a cylinder: LA = π·dh, where d is the diameter and h is the height
area of a rectangle: A = lw, where l is the length and w is the width
volume of a cone or pyramid: V = 1/3·Bh, where B is the area of the base and h is the height
volume of a cylinder or prism: V = Bh, where B is the area of the base and h is the height

You will notice that for lateral area purposes, a pyramid or cone is equivalent to a prism or cylinder of height equal to half the slant height. And for volume purposes, the volume of a pyramid or cone is equal to the volume of a prism or cylinder with the same base area and 1/3 the height.

Since the measurements are given in cm, we will use cm for linear dimensions, cm^2 for area, and cm^3 for volume.

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The heights of the cones at the top of the towers can be found from the Pythagorean theorem.

(slant height)^2 = (height)^2 + (radius)^2

height = √((slant height)^2 - (radius)^2) = √(10^2 -5^2) = √75 = 5√3

The heights of the pyramids can be found the same way.

height = √(13^2 -2^2) = √165

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<u>Area</u>

The total area of the castle will be ...

total castle area = castle lateral area + castle base area

These pieces of the total area are made up of sums of their own:

castle lateral area = cone lateral area + pyramid lateral area + cylinder lateral area + cutout prism lateral area

and ...

castle base area = cylinder base area + cutout prism base area

So, the pieces of area we need to find are ...

cone lateral area (2 identical cones)
pyramid lateral area (2 identical pyramids)
cylinder lateral area (3 cylinders, of which 2 are the same)
cutout prism lateral area
cylinder base area (3 cylinders of which 2 are the same)
cutout prism base area

Here we go ...

Based on the above discussion, we can add 1/2 the slant height of the cone to the height of the cylinder and figure the lateral area of both at once:

area of one cone and cylinder = π·10·(18 +10/2) = 230π

area of cylinder with no cone = top area + lateral area = π·1^2 +π·2·16 = 33π

area of one pyramid = 4·4·(13/2) = 52

The cutout prism outside face area is equivalent to the product of its base perimeter and its height, less the area of the rectangular cutouts at the top of the front and back, plus the area of the inside faces (both vertical and horizontal).

outside face area = 2((23+4)·11 -3·(23-8)) = 2(297 -45) = 504

inside face area = (3 +(23-8) +3)·4 = 84

So the lateral area of the castle is ...

castle lateral area = 2(230π + 52) +33π + 504 + 84 = 493π +692

≈ 2240.805 . . . . cm^2

The castle base area is the area of the 23×4 rectangle plus the areas of the three cylinder bases:

cylinder base area = 2(π·5^2) + π·1^2 = 51π

prism base area = 23·4 = 92

castle base area = 51π + 92 ≈ 252.221 . . . . cm^2

Total castle area = (2240.805 +252.221) cm^2 ≈ 2493.0 cm^2

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<u>Volume</u>

The total castle volume will be ...

total castle volume = castle cylinder volume + castle cone volume + castle pyramid volume + cutout prism volume

As we discussed above, we can combine the cone and cylinder volumes by using 1/3 the height of the cone.

volume of one castle cylinder and cone = π(5^2)(18 + (5√3)/3)

= 450π +125π/√3 ≈ 1640.442 . . . . cm^3

volume of flat-top cylinder = π·1^2·16 = 16π ≈ 50.265 . . . . cm^3

The volume of one pyramid is ...

(1/2)4^2·√165 = 8√165 ≈ 102.762 . . . . cm^3

The volume of the entire (non-cut-out) castle prism is the product of its base area and height:

non-cutout prism volume = (23·4)·11 = 1012 . . . . cm^3

The volume of the cutout is similarly the product of its dimensions:

cutout volume = (23 -8)·4·3 = 180 . . . . cm^3

so, the volume of the cutout prism is ...

cutout prism volume = non-cutout prism volume - cutout volume

= 1012 -180 = 832 . . . . cm^3

Then the total castle volume is ...

total castle volume = 2·(volume of one cylinder and cone) + (volume of flat-top cylinder) +2·(volume of one pyramid) +(cutout prism volume)

= 2(1640.442) + 50.265 +2(102.762) +832 ≈ 4368.7 . . . . cm^3

4 0

3 years ago

2 and square root of 2 + square root of 8 - 4 and square root of 1 over 2

Orlov [11]

Answer:

2.828427125, 2.82 or just round it to 3 if you want

Step-by-step explanation:

All you have to do is type it into the Desmos Scientific Calculator exactly how you see it.

3 0

3 years ago

TIME REMAINING

Norma-Jean [14]

Step-by-step explanation:

I am not sure I understand the answer options.

but normally the answer should be "commission".

as this is the defined way of payment, when you get paid based on the items sold.

but in any case, what math question is this ?

4 0

2 years ago

Not Answered Country Financial, a financial services company, uses surveys of adults age 18 and older to determine if personal f

ehidna [41]

Answer:

Null hypothesis: $p_{1} = p_{2}$

Alternative hypothesis: $p_{1} \neq p_{2}$

$z=\frac{0.410-0.350}{\sqrt{0.382(1-0.382)(\frac{1}{1000}+\frac{1}{900})}}=2.688$

$p_v =2*P(Z>2.688)= 0.0072$

Comparing the p value with the significance level assumed $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportions analyzed are significantly different at 5% of significance.

Step-by-step explanation:

Data given and notation

$X_{1}=410$ represent the number of people indicating that their financial security was more than fair in 2012

$X_{2}=315$ represent the number of people indicating that their financial security was more than fair in 2010

$n_{1}=1000$ sample 1 selected

$n_{2}=900$ sample 2 selected

$p_{1}=\frac{410}{1000}=0.410$ represent the proportion estimated of people indicating that their financial security was more than fair in 2012

$p_{2}=\frac{315}{900}=0.350$ represent the proportion estimated of people indicating that their financial security was more than fair in 2010

$\hat p$ represent the pooled estimate of p

z would represent the statistic (variable of interest)

$p_v$ represent the value for the test (variable of interest)

$\alpha=0.05$ significance level given

Part a: Concepts and formulas to use

We need to conduct a hypothesis in order to check if is there is a difference between the two proportions, the system of hypothesis would be:

Null hypothesis: $p_{1} = p_{2}$

Alternative hypothesis: $p_{1} \neq p_{2}$

Hypothesis testing

We need to apply a z test to compare proportions, and the statistic is given by:

$z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$ (1)

Where $\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{410+315}{1000+900}=0.382$

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

Calculate the statistic

Replacing in formula (1) the values obtained we got this:

$z=\frac{0.410-0.350}{\sqrt{0.382(1-0.382)(\frac{1}{1000}+\frac{1}{900})}}=2.688$

Statistical decision

Since is a two sided test the p value would be:

$p_v =2*P(Z>2.688)= 0.0072$

Comparing the p value with the significance level assumed $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportions analyzed are significantly different at 5% of significance.

3 0

3 years ago

I need Help ASAP!!! Please

Irina-Kira [14]

You can plug 1350 or less watts without overloading the circuit.

Step-by-step explanation:

Given,

Circuit overloads at = 2400 watts

Electricity consumed by portable heater = 1050 watts

Let,

x be the additional number of watts you can plug in.

Portable heater + additional watts ≤ Overloading limit

1050 + x ≤ 2400

Subtracting 1050 from both sides

$1050-1050+x\leq 2400-1050\\x\leq 1350$

You can plug 1350 or less watts without overloading the circuit.

Keywords: inequality, subtraction

Learn more about subtraction at:

#LearnwithBrainly

8 0

3 years ago

(3) Lines that forms straight angles​

(3) Lines that forms straight angles