Prove that the diagonals of a rectangle bisect each other.

The midpoint of BD is _____

[tex]\log_ab-\log_ac=\log_a\dfrac{b}{c}\\\\\log_315-\log_310=\log_3\dfrac{15}{10}=\boxed{\log_3\dfrac{3}{2}}[/tex]

Answer:

B.

Step-by-step explanation:

They have the same base so use the quotient rule (since they have a minus sign in between).

We are using this rule: [tex] log_a(b)-log_a(c)=log_a(\frac{b}{c})[/tex]

So we have  [tex] log_3(15)-log_3(10)=log_3(\frac{15}{10})[/tex]

We can reduce the inside fraction... 15/10=3/2

So the answer is B.

Answer:

10

Step-by-step explanation:

I am pretty sure this is the answer because, the two sides added have to be greater than the longest side.

Answer:

Step-by-step explanation:

We have

the rectangle 12ft × 6ft

two triangles with the base b = 12ft and the height h = 8ft

two triangles with the base b = 6ft and the height h = 9.5ft.

The formula of an area of a rectangle l × w:

[tex]A=lw[/tex]

Substitute:

[tex]A_1=(12)(6)=72\ dt^2[/tex]

The formula of an area of a triangle:

[tex]A=\dfrac{bh}{2}[/tex]

Substitute:

[tex]A_2=\dfrac{(12)(8)}{2}=48\ ft^2[/tex]

[tex]A_3=\dfrac{(6)(9.5)}{2}=28.5\ ft^2[/tex]

The Surface Area:

[tex]S.A.=A_1+2A_2+2A_3[/tex]

Substitute:

[tex]S.A.=72+2(48)+2(28.5)=225\ ft^2[/tex]

Answer:

Total surface area of the pyramid = 225 ft²

Step-by-step explanation:

Total surface area of the given pyramid is defined by  

(Area of rectangular base) + 2(area of two triangular sides with height 8 ft and base 12 ft) + 2(area of two triangles with height 9.5 ft and base 6 ft)

Total surface area = (12×6) + 2[[tex]\frac{1}{2}\times(h)(b)[/tex]]+2[[tex]\frac{1}{2}\times(h')(b')[/tex]]

= 72 + 2[[tex]\frac{1}{2}\times(8)(12)[/tex]]+2[[tex]\frac{1}{2}\times(9.5)(6)[/tex]]

= 72 + 96 + 57

= 225 ft²

Therefore, total surface area of the pyramid is 225 ft²

Answer:

[tex]10^4[/tex] or 10000

Step-by-step explanation:

So this is just a moving over of decimal.  It is a factor of 10. How many factors of 10 depends on many times we need to bring that decimal to the right.

Let's count.

3.4758.17

    | | | |

    That is 4 times to the right.

So you need 4 factors of 10, or [tex]10^4[/tex].

You need to multiply by [tex]10^4[/tex] (or 10000)

10,000

In your question, it asks what number you have to multiply by to get from 3.475817 to 34,758.17.

In order to find your answer, we can perform the quickest way to find the answer by moving the decimal place to the right.

To get from 3.475817 to 34,758.17, we're going to need to move the decimal place 4 times to the right, due to the fact that we're trying to get to a number where the decimal place is farther to the right.

We know that decimals goes by 10s, meaning that we would need to multiply by 10 in each move.

In other words, every time we move the decimal point to the right, we're multiplying by 10.

Since we moved 4 times, we need to multiply 10, 4 times.

[tex]10*10*10*10=10,000[/tex]

When you mutliply the 10s, you should get 10,000.

This means that you need to multiply by 10,000 to get from 3.475817 to 34,758.17

You can multiply 3.475817 by 10,000 to see if it gets you 34,758.17

[tex]3.475817 * 10000=34,758.17[/tex]

When you multiply the numbers, you would get 34,758.17, meaning that the answer is CORRECT.

Answer:

4/5

Step-by-step explanation:

The prime factorization of 24 is 2*2*2*3.

There prime factorization of 30 is 2*3*5.

They have 2*3 in common.

2*3=6.

So we can divide numerator (top) and denominator (bottom) by 6 and this will reduce the fraction into simplest form.

[tex]\frac{24}{30}=\frac{24 \div 6}{30 \div 6}=\frac{4}{5}[/tex]

The simplified version of the fraction 24/30 is 4/5

Fraction is a mathematical number by which we can describe a part of a whole item.

A fraction has two parts, the top part is called numerator & the bottom part is called denominator.

Example : [tex]\frac{5}{7}[/tex] , where 5 is the numerator & 7 is the denominator.

The given fraction is 24/30

The prime factorisation of 24 gives 2×2×2×3 , i.e. 2×2×(2×3)

The prime factorisation of 30 gives 2×3×5 , i.e. 5×(2×3)

Common in both 24 & 30 is 2×3, i.e. 6

So, we have to divide numerator & denominator of the given fraction by 6 to get the simplified form.

Hence, 24/30 = (24/6)/(30/6) = 4/5

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Answer:

[tex]f(a)=5-8a+15a^2[/tex]

[tex]f(a+h)=5-8a-8h+15a^2+30ah+15h^2[/tex]

[tex]\frac{f(a+h)-f(a)}{h}=-8+30a+15h[/tex]

Step-by-step explanation:

We are given [tex]f(x)=5-8x+15x^2[/tex].

We want to find [tex]f(a)[/tex] so we just replace the x there with a giving us:

[tex]f(a)=5-8a+15a^2[/tex].

We want to find [tex]f(a+h)[/tex] so we just replace x with (a+h) now giving us:

[tex]f(a+h)=5-8(a+h)+15(a+h)^2[/tex].

We will need to distribute and multiply things out here for later use so let's go ahead and do that:

[tex]f(a+h)=5-8(a+h)+15(a+h)^2[/tex]

[tex]f(a+h)=5-8a-8h+15(a+h)(a+h)[/tex]

[tex]f(a+h)=5-8a-8h+15(a^2+2ah+h^2)[/tex]

[tex]f(a+h)=5-8a-8h+15a^2+30ah+15h^2[/tex]

We want to find [tex]\frac{f(a+h)-f(a)}{h}[/tex] where h is not 0.

This requires the parts we found above:

[tex]\frac{f(a+h)-f(a)}{h}[/tex]

[tex]\frac{(5-8a-8h+15a^2+30ah+15h^2)-(5-8a+15a^2)}{h}[/tex]

There are some thing that will zero out (cancel out) in the numerator.

You have 5-8a+15a^2 in both parenthesis and you are subtracting so that part zero's out so you have this now:

[tex]\frac{-8h+30ah+15h^2}{h}[/tex]

Now you can divide h from top and bottom giving you:

[tex]\frac{-8+30a+15h}{1}[/tex]

[tex]-8+30a+15h[/tex]

Answer:

Problem: [tex]\frac{x-8}{x+11} \cdot \frac{x+8}{x-11}[/tex]

Answer: [tex]\frac{x^2-64}{x^2-121}[/tex]

Step-by-step explanation:

[tex]\frac{x-8}{x+11} \cdot \frac{x+8}{x-11}[/tex]

Writing as one fraction:

[tex]\frac{(x-8)(x+8)}{(x+11)(x-11)}[/tex]

Now before we continue, notice both of your bottom and top are in the form of (a-b)(a+b) or (a+b)(a-b) which is the same format.

That is, we are multiplying conjugates on top and bottom.

When multiplying conjugates, all you have to do it first and last.

For example:

[tex](a-b)(a+b)=a^2-b^2[/tex].

So your problem becomes this after the multiplication of conjugates:

[tex]\frac{x^2-64}{x^2-121}[/tex]

Answer:

[tex] \frac { ( x - 8 ) ( x + 8 ) } { ( x + 1 1 ) ( x - 1 1 ) } [/tex]

Step-by-step explanation:

We are to find the product of the rational expression below:

[tex] \frac { x - 8 } { x + 1 1 } \times \frac { x + 8 } { x - 1 1 } [/tex]

We are to multiply these terms by taking the LCM to get:

[tex] \frac { ( x - 8 ) ( x + 8 ) } { ( x + 1 1 ) ( x - 1 1 ) } [/tex]

Since the signs of all the terms are different so we cannot add them up.

Answer:

c=3t is the equation

18 cubic yards of concrete is what we need for a 6 inch thick bridge.

Step-by-step explanation:

If something directly varies, that means there is a constant, k, such that when you multiply it to one variable you always get the other variable.

That is for this particular problem we have c=kt where c is cubic yards and t is thickness in inches.

So we are given from the first sentence: 12 cubic yards and 4 inches thick which means we have:

12=k(4)

Divide both sides by 4:

12/4=k

Simplify:

3=k

So the equation is c=3t for any point (c,t) in this relation.

So we want to know many cubic yards of concrete (we want to know (c) so that Dru can build a 6 inch (t) thick bridge.

c=3(6)

c=18

18 cubic yards

Answer:

A.

Step-by-step explanation:

[tex]y=2x+4[/tex]

[tex]y=x^2+x-2[/tex]

Both equations are solved for y so I'm just going to substitute

the 1st  y  (the 2x+4) into the second equation's y.

[tex]2x+4=x^2+x-2[/tex]

I'm going to get everything on one side so I have 0=ax^2+bx+c.

Subtract 2x and subtract 4 on both sides:

[tex]0=x^2-x-6[/tex]

Since the coefficient of x^2 is 1, all we have to do is find two numbers that multiply to be -6 and add up to be -1.

These numbers are -3 and 2.

So the factored form of our equation is:

[tex]0=(x-3)(x+2)[/tex]

This means we have x-3=0 or x+2=0.

x-3=0

Add 3 on both sides:

x=3

x+2=0

Subtract 2 on both sides:

x=-2

So now we need to find y. I'm going to choose to use the easier equation:

y=2x+4.

If x=3, then y=2(3)+4=6+4=10.  The ordered pair (3,10) is a solution.

If x=-2, then y=2(-2)+4=-4+4=0.  The orded pair (-2,0) is a solution.

Answer:

BT is Congruent to CT

Step-by-step explanation:

Since these two triangles are said to be congruent, that means that their side lengths and angle measures are congruent as well. So, since BT corresponds to CT in the diagram, they must be congruent.

Answer:

BT≅CT

Step-by-step explanation:

If two triangles are congruent it means that they have the same size and shape, or that they mirror each other. Since the side, AT, is common to both triangles, the congruent triangles are mirrored in this case, which means that side CT in ∠CAT corresponds to side BT in ∠BAT.

Answer:

77.3 to the nearest tenth.

Step-by-step explanation:

The total of the scores on the set of 27 tests

= 78*27 = 2106.

After  2 more had joined the total score is 2106 + 69 + 66

= 2241.

So the new mean score = 2241 / 29

= 77.3.

Answer:

77.3

Step-by-step explanation:

Recall that the arithmetic mean is found as follows:

             sum of all data

a. m. = ---------------------------

             number of data

                                                                              sum

The mean score of 27 test results is thus 78 = --------

                                                                                27

Note that we can find the sum of all 27 of the original scores by multiplying 78 by 27:  2106.

Now we include two more scores.  Instead of the original sum, 2106, we have

                   2106 + 69 + 66

a. mean = ---------------------------

                              29

which comes out to 2241/29 = 77.3

Answer:

c.

Step-by-step explanation:

That is the only answer choice that makes the most sense.

±10 = √100; in this case, they want the non-negative value, so it is 10.

Answer: c. 100

Step-by-step explanation: √-81 and √-3 are not possibly unless complex numbers applied

√100 = 10 > 9 valid answer

√27 = 5.196 < 9 invalid answer

√3 = 1.732 < 9 invalid answer

√81 = 9 ≡ 9 invalid answer

Only c is the answer.

Hope it helped!

Answer:

B. 271 square units

Step-by-step explanation:

The area is the sum of 3 rectangles areas

step 1         12 * 19 = 228

step 2       3 * 3   = 9

step 3      2 * 17 = 34

Now, we add all the answers we got to get the area.

= 228 + 34 + 9 =   271 square units

The area of the polygon is:

A = 2 × 12 + 14(12+2) + 3(12 + 2 + 3)

A = 2 × 12 + 14 × 14 + 3 × 17

A = 24 + 196 + 51

A = 220 + 51

A = 271 cm²

Answer:

17.8 cm²

Step-by-step explanation:

Given 2 similar figures with linear ratio = a : b, then

ratio of areas = a² : b²

For the 2 similar cones

ratio of radii = 2 : 5, hence

ratio of areas = 2² : 5² = 4 : 25

Let x be the area of the smaller cone then by proportion

[tex]\frac{x}{4}[/tex] = [tex]\frac{111}{25}[/tex] ( cross- multiply )

25x = 444 ( divide both sides by 25 )

x = 17.76

Hence area of smaller cone = 17.8 cm² ( to the nearest tenth )

17.8 cm²

The total area of the surface of a 3-dimensional figure. The surface area of the solid object is the measure of the total area that the surface of the object occupies.

we must to course choose 3 dissimilar faces to capture length l, width w, or height h:

Given 2 similar figures with linear ratio = a: b, then

ratio of areas = a² : b²

For the 2 similar cones

ratio of radii = 2 : 5, hence

ratio of areas = 2² : 5² = 4 : 25

Let x be the area of the smaller cone then by proportion

[tex]\frac{x}{4} =\frac{111}{25}[/tex]

=  ( cross- multiply )

25x = 444 ( divide both sides by 25 )

x = 17.76

therefore an area of smaller cone = 17.8 cm² ( to nearest tenth )

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Answer:

5 = y

Step-by-step explanation:

As discussed in my video on my channel [Username: MATHEMATICS WIZARD], whenever you are dividing mixed numbers, fractions, etcetera, you multiply the first term by the divisor's multiplicative inverse [reciprocal].

Ex: 2 ÷ 1⁄11 → 2 × 11

I hope this helps, and as always, I am joyous to assist anytime.

Answer:

x = -7/2 and 3

Step-by-step explanation:

Use foil. Solve for x.

Answer:

The factors are (2x + 7)(x - 3) and the solutions are -3.5 and 3

Step-by-step explanation:

* Lets explain how to factor a trinomial in the form ax² ± bx ± c:

- Look at the c term first.  

# If the c term is a positive number, then its factors r , s  will both

  be positive or both be negative.

# a has two factors h and k

# The sum of c and a is b.

# The brackets are (hx ± r)(kx ± s) where a = hk , c = rs and b = rk + hs

# If the c term is a negative number, then either r or s will be negative,

   but not both.

# a has two factors h and k

# The difference of c and a is b.

# The brackets are (hx + r)(kx - s) where a = hk , c = rs and b = rk - hs

* Lets solve the problem

∵ The equation is 2x² + x - 21 = 0

∵ The general form of the equation is ax² + bx + c = 0

∴ a = 2 , b = 1 , c = -21

∵ c is negative

∴ its factors r and s have different sign

∵ a = 2

∵ The factors of a are h , k

∵ 2 = 2 × 1

∴ h = 2 and k = 1

∵ -21 = 7 × -3

∴ r = 7 and s = -3

∵ The brackets are (hx + r)(kx - s)

∴ 2x² + x - 21 = (2x + 7)(x - 3)

∵ 2x² + x - 21 = 0

∴ (2x + 7)(x - 3) = 0

- Equate each bracket by 0

∴ 2x + 7 = 0 ⇒ subtract 7 from both sides

∴ 2x = -7 ⇒ divide both sides by 2

∴ x = -7/2 = -3.5

- OR

∴ x - 3 = 0 ⇒ add 3 to both sides

∴ x = 3

∴ The solutions are -3.5 and 3

* The factors are (2x + 7)(x - 3) and the solutions are -3.5 and 3

Answer:

r ≤ 29, r-5, The sale price can be compared with the regular price, r-5 ≤ 24

Step-by-step explanation:

Amount to spend = $24

Regular price = r

Sale = $5

Sale Price = r-5

The regular price will be at most $5 more than the amount Roopesh has to spend.

The sale price will be $24 or less than that for Roopesh to afford.

Inequality for regular price :

r-5 ≤ 24

r ≤ 29

Therefore, the product Roopesh can afford is $29 or less than that.

What is the unknown? r ≤ 29

Which expression can represent the sale price? Sale price = r-5 (mentioned above)

Which comparison could be used? The sale price can be compared with the regular price

Which inequality represents the situation? r-5 ≤ 24

!!

Answer:

2, 10.

Smallest number = 20.

Step-by-step explanation:

It must have 2 as a factor because 4 is a factor,  and 2 is a factor of 4.

Also 4 *5 = 20 so 10 must be a factor.

The smallest number possible is 20.

The smallest number that has both 4 and 5 as factors is 20. In addition to 4 and 5, this number also has 1, 2, 10, and 20 as factors.

When a number has 4 and 5 as factors, this means that it must be a multiple of both 4 and 5. As a result, the smallest number this could be would be 20 (since 4*5 equals 20). Moreover, that given number will also have 1, 2, and 10 as factors. This is because 1 is a factor of every number, 2 is a factor of every even number, and 10 is a result of 2*5.

The number will also be divisible by 20 itself, because any number must be divisible by itself. This basic fact is often forgotten but very important.

To summarize, the factors of the smallest possible number would be 1, 2, 4, 5, 10, and 20.

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Final answer:

The 100th term of the arithmetic sequence is found using the formula an = a1 + (n-1)d. In this sequence, the 100th term is -290.

Explanation:

The question asks for the 100th term of an arithmetic sequence with a common difference. The given sequence is 7, 4, 1, -2, ... which shows a constant decrease of 3 between consecutive terms, hence it's an arithmetic sequence with a common difference (-3). To find the 100th term (a100) of the sequence, we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n-1)d, where a1 is the first term, and d is the common difference.

Plugging in the values, we get:

We calculate the 100th term as follows:

a100 = 7 + (100 - 1)(-3)

a100 = 7 - 297

a100 = -290

Therefore, the 100th term of the sequence is -290.

Answer:

7 lbs

Step-by-step explanation:

Given:

apples per pound= 88 cents

total money= $6.00

                   =600 cents

Now

1 lb= 88 cents

let x be the pounds of apple in 600 cents then

x lb=600 cents

By cross-multiplication we get:

600=88x

x=600/88

x= 6.81

x=7 pounds

7 pounds of apple can be bought by $6.00 !

Answer: 7 lb

Step-by-step explanation:

You know that in that fruit stand each pound of apple costs 88 cents.

Therefore, if you have $6.00, you need to divide this amount by 88 cents ($0.88 ) in order to calculate the pounds of apples you can buy.

Therefore, the pounds of apples you can buy to the nearest whole number is the following:

[tex]amount\ of\ apples=\frac{6.00}{0.88}=7\ lb[/tex]

Answer:

1. Rectangle

2. 376.8 cm

3. 379.94 square feet

4. –4 and 4

Step-by-step explanation:

A square is always a rectangle.

If a wheel has a radius of 15 cm, it would travel approximately 376.8 cm per 4 revolutions.

If the diameter of a circular garden is 22 feet, the approximate area of the garden is 379.94 square feet.

The solutions to the equation y2 – 1 = 15 is –4 and 4.

rectangle

376.8 cm

379.94 square feet

–4 and 4

Answer:

Yea, it is A

Step-by-step explanation:

Answer:

c) 325

Step-by-step explanation:

The fraction of students that went to the movies is 14/112. To find the number of students who went to the movies from a larger group, you have to find a fraction equal to 14/112, whose denominator is 2600 (the number of students from the larger group).

14/112 = x/2600

You can do cross multiplication here. Cross multiplication is multiplying the denominator of one fraction by the numerator of the fraction equivalent to it. That product will be equal to the numerator of the first fraction * denominator of the second fraction.

112*x = 14*2600

x = 325

Approximately 325 students out of 2600 students would've gone to the movie last weekend.

Answer:

60%

Step-by-step explanation:

To find percent change you take the new value, subtract it from the old value and then put that number over the original value. In this case it would be 960-600=360, 360/600=36/60=6/10 or 60 percent.

Answer:

D

Step-by-step explanation:

In the currency pair USD/CAN, USD is the base currency. The value of this base currency is always one unit. The relation between the pair demonstrates the concept that when one currency strengthens, the other weakens.

In the currency pair USD/CAN, USD is the base currency. It is then compared to the other currency, called the quote or counter currency, in this case, the Canadian dollar (CAN). The base currency is always worth one unit, and the exchange rate you see tells you how much of the counter currency it takes to buy one unit of the base currency. A fall in the Canada $/U.S. $ ratio means a rise in the U.S. $/Canada $ ratio, demonstrating the idea that the appreciation or strengthening of one currency often implies the depreciation or weakening of the other.

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In the currency pair USD/CAN, USD is the base currency.

Currency refers to a system of money used in a specific country or region, including coins and banknotes. It serves as a medium of exchange for goods and services. Currencies have different denominations and exchange rates, facilitating economic transactions within their respective jurisdictions.

In the currency pair USD/CAN, USD is the base currency. In a currency pair, the base currency represents the value of one unit of that currency in terms of the other currency. In this case, the USD/CAN currency pair means that 1 USD is equivalent to a certain amount of Canadian dollars. For example, if the exchange rate is 1 USD = 1.25 CAD, it means that 1 USD can be exchanged for 1.25 Canadian dollars.

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Answer:

Exponent.

[tex]\Huge \boxed{25}[/tex]

Step-by-step explanation:

2 is should be exponent.

5 is the base.

In the expression 5², the 2 represents the exponent.

[tex]\displaystyle 5^2=5\times5=25[/tex]

Exponent, and 25 is the correct answer.

The 2 in [tex]5^{2}[/tex] would represent the exponent.

An exponent is the number of times a base is multiplied by itself. The base on the other hand is the number that is being raised to a certain power.

Image is provided

Answer:

other answer is wrong this is the correct answer 100%

Step-by-step explanation:

x= 1

and

x= "n.a."

Answer:

90 yards

Step-by-step explanation:

If the length of a rectangle is four times it’s width and the area of the rectangle is 324 ft.², its perimeter is 90 yards.

Formula: Length = Area / Width

P = 2(4x) + 2(x)

P = 72 + 18

Its perimeter if 90 ft.

If length of the rectangle be l & width of the rectangle be w.

Then Area of the rectangle (A) = Length × Width

                                                   = (l × w) sq. unit

If length of the rectangle be l & width of the rectangle be w.

Then, Perimeter of the rectangle = 2×(l+w) unit

Given, Area of the rectangle = 324 sq. ft

Let, Width (w) = x unit

So, Length (l) = 4 times it's width = 4x unit

∴ Area = l × w = (4x)(x)

⇒ 324 = 4x²

⇒ x² = 324/4

⇒ x² = 81

⇒ x = [tex]\sqrt{}[/tex]81

⇒ x = 9  (neglecting negative value)

So, it's width = 9 ft.

Then, its length = (4×9) ft. = 36 ft.

∴ Perimeter =  2×(l+w) =  2×(36+9) ft. =  (2×45) ft. = 90ft.

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Answer:

$1440

Step-by-step explanation:

First find the the area with feet.

18*16=288

Now convert into sq. yards

288/3=96

Finally, multiply 96 by 15

96*15=1440

$1440

Answer:

$1440

Step-by-step explanation:

It will cost $1440 to carpet a rectangular floor 18 by 16 feet if the carpet costs $15 a square yard.

18 feet ⋅ 16 feet = 288

288 feet ÷ 3 = 96

96 ⋅ 15 = 1440

Answer:

c. 6 units

Step-by-step explanation:

The diameter of a circle with a circumference of 18.84956 is 6 units.

Use the formula: C=2πr

d=C

π=18.85

π≈6