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3 years ago

What is 2Log5(5x^3)+1/3log5(x^2+6 written as a single logarithm?

Mathematics

2 answers:

AveGali [126]3 years ago

5 0

An important rule of logs is a*log b = log b^a.

Thus, 2 (log to the base 5 of )(5x^3) = (log to the base 5 of ) (5x^3)^2, or

(log to the base 5 of ) (25x^6).

Next, (1/3) (log to the base 5 of ) (x^2+6) = (log to the base 5 of ) (x^2+6)^(1/3).

Here, the addition in the middle of the given expression indicates multiplication:

2Log5(5x^3)+1/3log5(x^2+6) = (log to the base 5 of ) { (5x^3)^2 * (x^2+6)^(1/3) }.

Here we've expressed the given log quantity as a single log.

Vikentia [17]3 years ago

4 0

Answer: B on Edge. log5 (25x^6) 3 sqrt x^2+8. Hope this helps!

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Solve using elimination. 4. 2x + 3y = 12 2x – y = 4 5. 5x – 2y = - 19 2x + 3y = 0

ANTONII [103]

I am attaching a document showing the steps.

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3 years ago

How do I round 0.3 to the nearest thousandth

Luda [366]

0.3 is 3 tenths.
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0.300

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3 years ago

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.

___

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

___

Area

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

___

Volume

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

Sarah is making flower bouquets that she will put in 10 vases. She will put 15 roses and some lilies in each vase.

AnnyKZ [126]

A is correct because the f represents the lilies and then you add the 15 because that is how much roses there are going to be. Lastly just multiply the ten that will give you the answer

3 0

3 years ago

Ingrid makes quilts in designs that follow a specific pattern. The first three designs are shown. In designs, the white blocks r

Gekata [30.6K]

The number of white and grey blocks in the quilts follows a sequence given by the Design number of the quilt

The numbers and patterns of the blocks are;

a. Patterns;

The height of the white blocks is the Design number, while the rows are one more than the design number

The height of the outer grey blocks is 2 blocks more than the Design number, while their rows are 3 blocks more than the Design number

b. The function is $\underline{p(n) = n \cdot (n + 1)}$

c. The function is $\underline{b(n) = (n + 2) \cdot (n + 3) - n \cdot (n + 1)}}$

d. $\underline{t(n) = (n + 2) \cdot (n + 3) = b(n) - p(n) }$

e. The design chosen is Design 9

The number of blocks is 132 blocks

Reasons:

The pattern of the design are;

a. The column height of the grey blocks = 2 blocks more than the column height of the white blocks

Width of the outer grey blocks = 2 blocks more than the width of the inner white blocks

The number of white blocks in a design, n is the product of n and (n + 1)

The number of grey blocks in a design n, is the product of (n + 2) and (n + 3) less the number of white blocks in the design

b. Based on the above, the function that represent the number of picture blocks in Design n, is therefore;

p(n) = n·(n + 1)

c. Based on the above, the function that represent the number of border blocks in Design n, is therefore;

b(n) = (n + 2)·(n + 3) - n·(n + 1)

d. The total number of blocks, t(n) = (n + 2)·(n + 3)

t(n) = (n + 2)·(n + 3) - n·(n + 1) + n·(n + 1)

t(n) = p(n) + b(n)

e. The given number of picture blocks = 90

The number of picture blocks in a design, n = n × (n + 1)

Therefore, given that the number of picture blocks is 90, we have;

90 = n × (n + 1)

n² + n - 90 = 0

(n + 10)·(n - 9) = 0

n = 9, or n = -10

Therefore, the design museum chooses is Design 9

The total number of blocks in a quilt design, t(n) = (n + 2)·(n + 3)

Total blocks in the design the art museum choses, t(9), is given as follows;

t(9) = (9 + 2)·(9 + 3) = 132 blocks

Learn more about series here;

brainly.com/question/13151093

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2 years ago