MATHEMATICS
HIGH SCHOOL

Answer: A^5b -a

= a* (a^(5b-1) -1) (distributive property)

Now try backwards:

a* (a^(5b-1) -1)

= a* a^(5b-1) -a

= a^1* a^(5b-1) -a

= a^(5b-1+1) -a

= a^(5b) -a (correct).

The final answer is a* (a^(5b-1) -1)~

= a* (a^(5b-1) -1) (distributive property)

Now try backwards:

a* (a^(5b-1) -1)

= a* a^(5b-1) -a

= a^1* a^(5b-1) -a

= a^(5b-1+1) -a

= a^(5b) -a (correct).

The final answer is a* (a^(5b-1) -1)~

MIDDLE SCHOOL

How to find the surface area of certain 3 dimensional shapes

There are formulas for the area of triangles, various quadrilaterals, regular polygons, sections of circles and spheres (including whole circles and spheres). In general, you make use of the appropriate formula or combination of formulas for the shape of interest.

If the shape does not conform to one of these, but is described by a formula, then methods of calculus (integration) can be used to find the surface area.

A few of the relevant formulas are

.. triangle area = (1/2)bh . . . . b = base length; h = height measured perpendicular to the base

.. rectangle or parallelogram area = bh . . . . b and h defined as for a triangle

.. trapezoid area = (1/2)(b₁ +b₂)h . . . . b₁ and b₂ are the lengths of the parallel bases, h is the perpendicular distance between them

.. circle area = π*r^2 . . . . r is the radius

.. sphere area = 4π*r^2 . . . . r is the radius

.. lateral area of a cone = π*rh . . . . r is the radius of the base, h is the slant height

If the shape does not conform to one of these, but is described by a formula, then methods of calculus (integration) can be used to find the surface area.

A few of the relevant formulas are

.. triangle area = (1/2)bh . . . . b = base length; h = height measured perpendicular to the base

.. rectangle or parallelogram area = bh . . . . b and h defined as for a triangle

.. trapezoid area = (1/2)(b₁ +b₂)h . . . . b₁ and b₂ are the lengths of the parallel bases, h is the perpendicular distance between them

.. circle area = π*r^2 . . . . r is the radius

.. sphere area = 4π*r^2 . . . . r is the radius

.. lateral area of a cone = π*rh . . . . r is the radius of the base, h is the slant height

HIGH SCHOOL

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.

V = x · y²

2 x + y = 3

y = 3 - 2 x

V = x · ( 3 - 2 x )² = x · ( 9 - 12 x + 4 x² ) = 9 x - 12 x² + 4 x³

V ` = 9 - 24 x + 12 x² = 3 ( 4 x² - 8 x + 3 ) = 3 ( 4 x² - 6 x - 2 x + 3 ) =

= 3 [ 2 x ( 2 x - 3 ) - ( 2 x - 3 ) ] = 3 ( 2 x - 3 ) ( 2 x - 1 )

The largest volume is when: V ` = 0, so:

2 x - 3 = 0

2 x = 3

x = 1.5 ( which is incorrect, because : y = 0 )

or: 2 x - 1 = 0

2 x = 1

x = 0.5, y = 3 - 2 · 0.5 = 3 - 1 = 2

V max = 0.5 · 2² = 0.5 · 4 = 2 ft³

2 x + y = 3

y = 3 - 2 x

V = x · ( 3 - 2 x )² = x · ( 9 - 12 x + 4 x² ) = 9 x - 12 x² + 4 x³

V ` = 9 - 24 x + 12 x² = 3 ( 4 x² - 8 x + 3 ) = 3 ( 4 x² - 6 x - 2 x + 3 ) =

= 3 [ 2 x ( 2 x - 3 ) - ( 2 x - 3 ) ] = 3 ( 2 x - 3 ) ( 2 x - 1 )

The largest volume is when: V ` = 0, so:

2 x - 3 = 0

2 x = 3

x = 1.5 ( which is incorrect, because : y = 0 )

or: 2 x - 1 = 0

2 x = 1

x = 0.5, y = 3 - 2 · 0.5 = 3 - 1 = 2

V max = 0.5 · 2² = 0.5 · 4 = 2 ft³

MIDDLE SCHOOL

Last month 5,389 people shopped at the bookstore .What is this number rounded to the nearest hundred?

5, 400 because 389 is closer to 400 than 300

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Glad I could help, and good luck!

____________________________________

Glad I could help, and good luck!

MIDDLE SCHOOL

A 3ft vertical post casts a 24in shadow at the same time a pine tree casts a 30ft shadow. How tall is the pine tree?

The ratio of the height and shadow will be constant. Then the height of the pine tree will be 3.75 feet.

What are ratio and proportion?A ratio is a collection of ordered integers a and b represented as a/b, with b never equaling zero. A proportionate expression is one in which two items are equal.

A 3 feet vertical post casts a 24 in shadow at the same time a pine tree casts a 30 feet shadow.

The ratio of the height and shadow will be constant.

Then the height of the pine tree will be

Let x be the height of the pine tree.

x / 30 = 3 / 24

x = 3.75 feet

More about the ratio and the proportion link is given below.

brainly.com/question/14335762

#SPJ2

The pine tree is 45ft tall