Please help it’s for tmr, I only have 0 minute left
Leo has a number of toy soldiers between 27 and 54. If he wants to group them four by four, there are none left, seven by seven, 6 remain, five by five, 3 remain. How many toy soldiers are there?
The answer is 48 but I need step by step explanation

Answers

Answer 1

Leo has 48 toy soldiers.

What is the key concept is used to solve the question ?

This problem involves finding a number that satisfies certain conditions related to division with remainders. Specifically, the number must be divisible by 4 with no remainder, must leave a remainder of 6 when divided by 7, and must leave a remainder of 3 when divided by 5.

Calculating the number of toy soldiers :

Let's start by using the first condition to narrow down the possibilities for the number of toy soldiers. We know that the number must be divisible by 4 with no remainder, and it must be between 27 and 54. The multiples of 4 in this range are 28, 32, 36, 40, 44, 48, and 52.

Next, we can use the second condition to eliminate some of these possibilities. If we divide each of these numbers by 7, we get the following remainders:

28 ÷ 7 = 4 remainder 0

32 ÷ 7 = 4 remainder 4

36 ÷ 7 = 5 remainder 1

40 ÷ 7 = 5 remainder 5

44 ÷ 7 = 6 remainder 2

48 ÷ 7 = 6 remainder 6

52 ÷ 7 = 7 remainder 3

The only number in our list that leaves a remainder of 6 when divided by 7 is 48, so we can tentatively conclude that this is the number of toy soldiers.

Finally, we can check the third condition to make sure that 48 leaves a remainder of 3 when divided by 5. Indeed, 48 ÷ 5 = 9 remainder 3, so all of the conditions are satisfied.

Therefore, we can confidently say that Leo has 48 toy soldiers.

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Related Questions

the quadratic sequence: 44; 52; 64; 80; Write down the next two terms of the sequence. Determine the nth term of the quadratic sequence. Calculate the 30th term of the sequence. Prove that the quadratic sequence will always have even terms.​

Answers

To find the next two terms of the sequence, we need to first find the common difference between consecutive terms:

52 - 44 = 8

64 - 52 = 12

80 - 64 = 16

We notice that the common difference is increasing by 4 for each term. Therefore, the next two terms of the sequence are:

80 + 20 = 100

100 + 24 = 124

To determine the nth term of the quadratic sequence, we can use the formula:

an = a1 + (n-1)d + bn^2

where a1 is the first term, d is the common difference, b is the coefficient of n^2, and n is the term number.

Using the first four terms of the sequence, we can form a system of equations:

44 = a1 + b

52 = a1 + d + b

64 = a1 + 2d + b

80 = a1 + 3d + b

Solving for a1 and b, we get:

a1 = 20

b = 24

Substituting these values into the formula for an, we get:

an = 20 + (n-1)4 + 24n^2

an = 24n^2 + 4n - 4

To find the 30th term of the sequence, we simply substitute n = 30 into the formula we just derived:

a30 = 24(30)^2 + 4(30) - 4

a30 = 21,596

To prove that the quadratic sequence will always have even terms, we notice that the first term is even (44 = 2 x 22), and the common difference is even (8 = 2 x 4). Therefore, every term of the sequence can be expressed as an even number plus an even multiple of n^2, which is always even. Hence, the quadratic sequence will always have even terms.

Step-by-step explanation:

Sequence is 44;52;64;80;.....44;52;64;80;.....

General formula is Tn=2n2+2n+40

Many bank accounts never go below zero. But some banks will allow a negative balance, at least for a short time, called an overdraft. It means someone has taken out, or 'drafted', more money than was in the account to begin with. Jose's account has gone into overdraft. His balance is $-27.14. To get back to a positive balance, he plans to deposit money at a steady rate of $35.03 per week. How much will be in his account after 7 weeks?

Answers

Answer:

yo your dûmb asl

Step-by-step explanation:

Masons backyard deck is rectangular. The width is 12 feet less than the length. The perimeter is 64 feet. What is the length?

Answers

As Masons backyard deck is rectangular, the length of the deck is 22 feet.

Let's start by using algebra to solve for the length of the rectangular deck.

Let L be the length of the deck.

Then, the width of the deck is L - 12.

The perimeter is the sum of all four sides, so we have:

Perimeter = 2L + 2(L - 12) = 64

Simplifying the equation, we get:

2L + 2L - 24 = 64

Combining like terms, we get:

4L - 24 = 64

Adding 24 to both sides, we get:

4L = 88

Dividing both sides by 4, we get:

L = 22

Therefore, the length of the deck is 22 feet.

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xavier is a teacher and takes home 90 papers to grade over the weekend. he can grade at a rate of 6 papers per hour. how many papers would xavier have remaining to grade after working for 12 hours?

Answers

The number of papers xavier have remaining after working for 12 hours is 18

How many papers would xavier have remaining

Xavier can grade 6 papers per hour, so in 12 hours he can grade:

6 papers/hour x 12 hours = 72 papers

Therefore, after working for 12 hours, Xavier would have

90 - 72 = 18 papers remaining to grade.

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In 915. 23, the digit 3 is in the

place.

Answers

Answer:

hundreth

Step-by-step explanation:

the 2 is in the tenth and the 3 is in the hundreth

Find the definite integral of f(x)=
fraction numerator 1 over denominator x squared plus 10 invisible times x plus 25 end fraction for x∈[
5,7]

Answers

Over the range [5, 7], the definite integral of f(x) = 1 / (x² + 10x + 25) is around -1/60.

To find the definite integral of f(x) = 1 / (x² + 10x + 25) over the interval [5, 7], we can use the following formula:

∫[a,b] f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

First, we need to find the antiderivative of f(x):

∫ f(x) dx = ∫ 1 / (x² + 10x + 25) dx

To do this, we can use a technique called partial fraction decomposition:

1 / (x² + 10x + 25)

= A / (x + 5) + B / (x + 5)²

Multiplying both sides by the denominator (x² + 10x + 25), we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

A + B = 1

A + 1 = 1

A = 0

Therefore, the partial fraction decomposition of f(x) is:

1 / (x² + 10x + 25) = 1 / (x + 5)²

Now we can find the antiderivative:

∫ f(x) dx = ∫ 1 / (x² + 10x + 25) dx = ∫ 1 / (x + 5)² dx

Using the substitution u = x + 5, du = dx, we get:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

where C is the constant of integration.

Now we can evaluate the definite integral over the interval [5, 7]:

∫[5,7] f(x) dx = F(7) - F(5)

∫[5,7] f(x) dx = [-1 / (7 + 5) + C] - [-1 / (5 + 5) + C]

∫[5,7] f(x) dx = [-1 / 12 + C] - [-1 / 10 + C]

∫[5,7] f(x) dx = -1 / 12 + C + 1 / 10 - C

∫[5,7] f(x) dx = -1 / 60

Therefore, the definite integral of f(x) = 1 / (x² + 10x + 25) over the interval [5, 7] is approximately -1/60.

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andrew is buying a cell phone that has a regular price of $485. the cell phone is on sale for 35% off the regular price. what will be the sale price?

Answers

the sale price of the cell phone after the 35% discount is $315.25.

How to solve and what is sale?

To find the sale price of the cell phone, we need to apply the discount of 35% to the regular price of $485. We can do this by multiplying the regular price by 0.35 and then subtracting the result from the regular price:

Sale price = Regular price - Discount amount

Sale price = $485 - (0.35 x $485)

Sale price = $485 - $169.75

Sale price = $315.25

Therefore, the sale price of the cell phone after the 35% discount is $315.25.

A sale is a temporary reduction in the price of a product or service. Sales are often used by businesses to attract customers and increase sales volume. Sales can be offered for many reasons, such as to clear out inventory, promote a new product, or attract customers during a slow period.

In a sale, the price of a product or service is discounted, either by a fixed amount or by a percentage of the regular price. For example, a store might offer a 20% discount on all clothing items, or a car dealership might offer a $5,000 discount on a particular model of car.

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plsss help theorical probability

calculate the theoretical probability of a 1 eyed, 1 horned, flying, purple, people eater

Answers

The theoretical probability for the 1 horned, 1 eyed,  flying, people eater purple is found to be 1/120.

Explain about the theoretical probability?

Experimental Probability: Based on actual results rather than mathematical calculations, the experimental probability of an occurrence is the likelihood that the event will actually occur.

Theoretical Probability: Considering that the event is ideal, the theoretical chance that it will occur is the theoretically ideal probability of a specific result. The flaws in the system are not taken into consideration by theoretical probability.

Theoretical Probability = Number of favorable outcomes / Number of possible outcomes.

The given probability are;

1 eyed - 3/41 horned - 1/5flying - 2/3 purple -3/8 people eater - 1/2

Let P(E)  be the theoretical probability  1 horned, 1 eyed,  flying, people eater purple.

Then,

P(E) =  3/4 * 1/5* 2/3* 3/8* 1/2

P(E) =  1/120

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For both f(x)= √x and f(x)=1/x, sketch the graph of the parent function, apply the transformations indicated, and state the domain and range. Note: You can sketch the graphs by hand or in digital form.

a) y= f(x+2)-1
b) y= -2f(x)+4
c) y= -2f(-(x-3))+1

Answers

Answer: a) Parent function:

f(x) = √x

Domain: x ≥ 0

Range: y ≥ 0

Applying transformations:

shift 2 units left: f(x+2)

shift 1 unit down: f(x+2)-1

Final equation and graph:

y = √(x+2) - 1

Domain: x ≥ -2

Range: y ≥ -1

b) Parent function:

f(x) = 1/x

Domain: x ≠ 0

Range: y ≠ 0

Applying transformations:

multiply by -2: -2f(x)

shift 4 units up: -2f(x)+4

Final equation and graph:

y = -2/x + 4

Domain: x ≠ 0

Range: y ≠ 4

c) Parent function:

f(x) = 1/x

Domain: x ≠ 0

Range: y ≠ 0

Applying transformations:

shift 3 units right: f(-(x-3))

multiply by -2: -2f(-(x-3))

shift 1 unit up: -2f(-(x-3))+1

Final equation and graph:

y = -2/(3-x) + 1

Domain: x ≠ 3

Range: y ≠ 1

Step-by-step explanation:

h(x)= -x + 5, solve for x when h(x) = 3

Answers

According to the given information, the solution to H(x) = 3 is x = 2.

What is equation?

In mathematics, an equation is a statement that asserts the equality of two expressions. An equation typically consists of two parts: the left-hand side (LHS) and the right-hand side (RHS). The LHS and RHS are separated by an equals sign (=), indicating that they have the same value. The general form of an equation is: LHS = RHS

To solve for x when H(x) = 3, we substitute 3 for H(x) in the equation and solve for x:

H(x) = -x + 5

3 = -x + 5

Subtracting 5 from both sides, we get:

-2 = -x

Multiplying both sides by -1, we get:

2 = x

Therefore, the solution to H(x) = 3 is x = 2.

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[tex]\huge\text{Hey there!}[/tex]


[tex]\mathtt{h(x) = -x + 5}\\\\\mathtt{3 = -x + 5}\\\\\mathtt{-x + 5 = 3}\\\\\textsf{SUBTRACT 5 to BOTH SIDES}\\\\\mathtt{-x + 5 - 5 = 3 - 5}\\\\\textsf{SIMPLIFY it}\\\\\mathtt{-x = 3 - 5}\\\\\mathtt{-x = -2}\\\\\mathtt{-1x = -2}\\\\\textsf{DIVIDE }\mathsf{-1}\textsf{ to BOTH SIDES}\\\\\mathtt{\dfrac{-1x}{-1} = \dfrac{-2}{-1}}\\\\\textsf{SIMPLIFY it}\\\\\mathtt{x = \dfrac{-2}{-1}}\\\\\mathtt{x = 2}[/tex]


[tex]\huge\text{Therefore your answer should be:}\\\\\huge\boxed{\mathtt{x = 2}}\huge\checkmark[/tex]


[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]



~[tex]\frak{Amphitrite1040:)}[/tex]

What is the Area and Volume of this prism?

Answers

The area of the prism is given as follows: 95 m².The volume of the prism is given as follows: 150 m³.

How to calculate the area and the volume of a prism?

The area of the prism is given by the sum of the areas of all the parts that compose the prism.

The parts that compose the prism are given as follows:

Rectangular base of dimensions 3m and 5m.Rectangle of dimensions 5m and 10 m.Two right triangles of sides 3m and 10 m.

Hence the area of the prism is given as follows:

A = 3 x 5 + 5 x 10 + 2 x 1/2 x 3 x 10

A = 95 m².

The volume is given by the multiplication of the base area by the height, hence:

Base area = 5 x 3 = 15 m².Height of 10 m.

Thus:

V = 15 x 10 = 150 m³.

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mai has a jar of quarters and dimes. she takes at least 10 coins out of the jar and has less than $2.00. write a system of inequalities that represents the number of quarters, `x`, and the number of dimes, `y`, that mai could have.

Answers

The system of inequalities that represents the number of quarters, x, and the number of dimes, y, that Mai could have is given by:

x + y ≥ 10 and 0.25x + 0.1y < 2

These are the two systems of inequalities that represent the number of quarters, x, and the number of dimes, y, that Mai could have.

Let x be the number of quarters and y be the number of dimes that Mai has. Then, the system of inequalities can be represented as:

Thus, the first inequality is x + y ≥ 10.

Also, Mai has less than $2.00, therefore, the second inequality is 0.25x + 0.1y < 2. The value of x and y are assumed to be non-negative integers.+

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y=x^2+7x-3
complete the square to re-write the quadratic function in vertex form.
pls help

Answers

Answer:

Y=x^2+7x-3

complete the square to re-write the quadratic function in vertex form.

pls help

Step-by-step explanation:

To complete the square, we need to add and subtract a constant term inside the parentheses, which when combined with the quadratic term will give us a perfect square trinomial.

y = x^2 + 7x - 3

y = (x^2 + 7x + ?) - ? - 3 (adding and subtracting the same constant)

y = (x^2 + 7x + (7/2)^2) - (7/2)^2 - 3 (the constant we need to add is half of the coefficient of the x-term squared)

y = (x + 7/2)^2 - 49/4 - 3

y = (x + 7/2)^2 - 61/4

So the quadratic function in vertex form is y = (x + 7/2)^2 - 61/4, which has a vertex at (-7/2, -61/4).

Find the center of mass of a thin plate of constant density delta covering the given region. The region bounded by the parabola y = 3x - x^2 and the line y = -3x The center of mass is. (Type an ordered pair.)

Answers

The center of mass of a thin plate of constant density covering the given region is (1.8, 3.6).

To find the center of mass, we must calculate the weighted average of all the points in the region. The region is bounded by the parabola y = 3x - x² and the line y = -3x.

We must calculate the integral of the region and divide by the total mass. The mass is equal to the area times the density, .

The integral of the region is calculated using the limits of the two curves, yielding a final integral of 32/15. Dividing this integral by the density gives the total mass, and multiplying by the density gives us the center of mass, (1.8, 3.6).

We can also find the center of mass by calculating the moments of the plate about the x-axis and y-axis.

The moment about the x-axis is calculated by finding the integral of the parabola and line using the x-coordinate, and the moment about the y-axis is calculated by finding the integral of the parabola and line using the y-coordinate. Once the moments are found, we can divide each moment by the total mass to get the center of mass.

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Please please please help me!!!!!​

Answers

The volume of the sphere which is equivalent to the lung capacity is approximately =2,571 cm³

How to calculate the volume of the sphere?

To calculate the volume of a sphere the formula used = V = 4/3 πr³

Radius = 8.5 cm

First cube the radius = 8.5³ = 614.125

The, multiply r³ by π = r³×π = 614.125× 3.14= 1928.3525

Take this answer and multiply it by 4 = 4×1928.3525= 7713.41

Last, divide this answer by 3 = 7713.41/3 = 2571.136666

Therefore the volume of the balloon = 2,571 cm³(approximately)

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use the unique factorization theorem to write the following integers in standard factored form. (a) 504 (b) 819 (c) 5,445

Answers

Using the Unique factorization theorem for the following integers the standard factored form of 504 is 2³ x 3²x 7 , for 819 is 3² ×7×13 and for  5,445 is 3²×5×7².

The Unique Factorization Theorem states that any positive integer can be written as a product of prime numbers in a unique way. To write each of the integers in standard factored form.

Using this theorem, we can factorize any positive integer into its prime factors. Here are the steps to factorize a number:

   Find the smallest prime factor of the number.    Divide the number by this prime factor, and repeat step 1 with the result.    Continue this process until the result is 1.

The prime factors obtained in this process can then be multiplied together to obtain the standard factored form of the original number . Therefore,

)504 = 2³ x 3² x 7)819 = 3² ×7×13)5,445 =3²×5×7²

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MR. Swanson wants to buy some mugs as gifts on his trip to California.There are three gifts shops, and each is offering a different deal. Which gift shop has the best deal for mugs

Answers

Answer: The one that has the best deals.

Step-by-step explanation:

on saturday a local hamburger shop sold a combined total of 416 hamburgers and cheeseburgers.the number of cheeseburgers sold was three times the number of hamburgers sold. how many hamburgers were sold?

Answers

Answer: Let x be the number of hamburgers sold.

Then, the number of cheeseburgers sold is 3x.

The total number of burgers sold is x + 3x = 4x.

Given that the total number of burgers sold is 416, we have:

4x = 416

x = 416/4

x = 104

Therefore, 104 hamburgers were sold.

Step-by-step explanation:

Given ΔABC with measure of angle B equals 78 degrees, measure of angle C equals 52 degrees, and a = 16 inches, what is the length of b?

Answers

To find the length of side b in triangle ABC, we can use the Law of Sines. The length of side b is approximately 20.058 inches.

To find the length of side b in triangle ABC, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle.

Using the Law of Sines, we have:

sin(A)/a = sin(B)/b

We are given the measure of angle B as 78 degrees and side a as 16 inches. We can substitute these values into the equation:

sin(A)/16 = sin(78)/b

To find sin(A), we can use the fact that the sum of the angles in a triangle is 180 degrees:

A + B + C = 180

A + 78 + 52 = 180

A = 180 - 78 - 52

A = 50 degrees

Now we can substitute the values into the equation again:

sin(50)/16 = sin(78)/b

To solve for b, we can cross-multiply and isolate b:

b = (16 * sin(78))/sin(50)

We can calculate the length of side b by evaluating this expression using a calculator. The measurement will be roughly 20.058 inches.

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Y=5x+17 Y=-2x+4 solve with elimination

Answers

Answer:

x = -13/7

y = 54/7

Step-by-step explanation:

Y = 5x + 17                          Y = -2x + 4

5x + 17 = -2x + 4

7x + 17 = 4

7x = -13

x = -13/7

Not put -13/7 in for x and solve for y

y = 5(-13/7) + 17

y = 54/7

So, the answer is x = -13/7 and y = 54/7

Answer:   x = -13 / 7, y = 54/7

Step-by-step explanation:

To eliminate a variable, we can substitute y for 5x + 17

We get 5x + 17 = -2x + 4

7x = -13

x = -13 / 7

Substituting x into the 2nd equation y = 5 * -13 / 7 + 17

y = 119/7 - 65/7

y = 54/7

Simplify to an expression involving a single trigonometric function with no fractions.


cos(−x)+tan(−x)sin(−x)

Answers

Sec x is the simplified expression cos(−x)+tan(−x)sin(−x) involving a single trigonometric function with no fractions.

The functions of an angle in a triangle are known as trigonometric functions, commonly referred to as circular functions. In other words, these trig functions provide the relationship between a triangle's angles and sides. There are five fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

The Given expression is

cos(−x)+tan(−x)sin(−x)

Now,

cos(−x) + tan(−x)sin(−x)

= cos x + (- tan x) (- sin x)

= cos x + tan x * sin x

= cos x + (sin x / cos x) * sin x

= (cos²x + sin²x) / cos x     ( As sin²x + cos²x = 1)

= 1/ cos x

= sec x       (As sec x = 1/cos x)

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Determine whether the following subsets are subspaces of the given vector spaces or not.text Is end text W subscript 2 equals open curly brackets space p equals a subscript 2 t squared plus a subscript 1 t plus a subscript 0 space element of space straight double-struck capital p subscript 2 space left enclose space a subscript 0 equals 2 space end enclose close curly brackets space space text a subspace of the vector space end text space straight double-struck capital p subscript 2 ?(Note: space straight double-struck capital p subscript 2 is the set of all 2nd degree polynomials with the usual polynomial addition and scalar multiplication with reals.)Answer 1text Is end text W subscript 1 equals open curly brackets open square brackets table row a b c row d 0 0 end table close square brackets space element of space M subscript 2 x 3 space end subscript space left enclose space b equals a plus c space end enclose close curly brackets space text a subspace of the vector space end text space space M subscript 2 x 3 space end subscript?(Note: space M subscript 2 x 3 space end subscript is the set of all 2x3 matrices with the standart matrix addition and scalar multiplication with reals.)

Answers

Yes, W_2 = {p_2 = a_2t_2 + a_1t + a_0 ∈ ℙ_2 | a_0 = 2} is a subspace of the vector space ℙ_2.

Yes, W_1 = {[a b c; d 0 0] ∈ M_{2x3} | b = a + c} is a subspace of the vector space M_{2x3}.

Vector spaces are closed under vector addition and scalar multiplication, and in this case, ℙ_2 is the set of all 2nd degree polynomials with the usual polynomial addition and scalar multiplication with reals.

Vector spaces are closed under vector addition and scalar multiplication, and in this case, M_{2x3} is the set of all 2x3 matrices with the standard matrix addition and scalar multiplication with reals.

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WILL GIVE BRAINLIEST 15 POINTS PLEASEE Fill in the blanks pleaseee

Answers

Therefore, we have the values of:

a = -g(x) for -10 < x < -8

b = lower limit of the range where g(x) = -6

c = -C for -1 < x < 1

d = upper limit of the range where g(x) = 4

e = we cannot determine the value of e based on the given information.

What is function?

In mathematics, a function is a rule that assigns a unique output value for every input value in its domain. It is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. Functions are often represented by a formula or an equation, but they can also be defined in other ways, such as through graphs, tables, or verbal descriptions. They are used to model a wide variety of phenomena in science, engineering, economics, and many other fields.

Here,

We can find the values of a, b, c, d, and e by examining the given information:

For -15 < x < -10: g(x) = -(-10) = 10

For -10 < x < -8: g(x) = -a

For -1 < x < 1: g(x) = -C

For b < x < l: g(x) = -(-6) = 6

For 10 < x < 15: g(x) = -8

For d < x < e: the value of g(x) is not specified in the given information, so we cannot determine the value of e based on this.

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I need help on 10 and 11

Answers

Answer:

x = +8

x = -8

That's the thing that I only know. Im a seventh grader(turned 13) so I wouldn't know this fully. Ask someone else to get it fully.

Step-by-step explanation:

5x + 6x - 10 + 12 = 90

11x = -88

x = -8

or

x = 8

If a car runs at a constant speed and takes 3 hrs to run a distance of 180 km, what time it
will take to run 100 km?

Answers

Answer:

100 minutes

Step-by-step explanation:

We know

It takes 3 hrs to run a distance of 180 km.

180 / 3 = 60 km / h

60 minutes = 60 km

40 minutes = 40 km

What time it will take to run 100 km?

60 + 40 = 100 minutes

So, it takes 100 minutes to run 100 km.

Find the radius of the sphere with the given volume

V=4500 mm^3

Answers

Answer:

10.24

Step-by-step explanation:

i used an online calculator

Select the correct answer. Which graph represents this equation? A. The graph shows an upward parabola with vertex (minus 3, minus 4.5) and passes through (minus 7, 3.5), (minus 6, 0), (0, 0), and (1, 3.5) B. The graph shows an upward parabola with vertex (3, minus 4.5) and passes through (minus 1, 3.5), (0, 0), (6, 0), and (7, 3.5) C. The graph shows an upward parabola with vertex (minus 2, minus 6) and passes through (minus 5, 7), (minus 4, 0), (0, 0), and (1, 7) D. The graph shows an upward parabola with vertex (2, minus 6) and passes through (minus 1, 7), (0, 0), (4, 0), and (5, 7)

Answers

Answer:

A

Step-by-step explanation:

Find the missing side of each triangle round your answers to the nearest 10th

Answers

ABC is a right angled triangle where perpendicular(p)= 8cm, Hypotenuse(h)= 10cm and base(b)= x
Now,
Base= root of h2-p2

Find the area under the curve y = 2 x^-3 from x = 6 to x = t and evaluate it for t = 10 , t = 100 . Then find the total area under this curve for x ≥ 6 .
(a) t = 10
(b) t = 100
(c) Total area

Answers

The total area under the curve for x ≥ 6 is 449/4500.

The area under the curve y = 2x-3 from x = 6 to x = t and its evaluation at t = 10 and t = 100The area under the curve y = 2x-3 from x = 6 to x = t can be calculated as follows:

We know that the area of the region under the curve f(x) between x = a and x = b is given by [tex]A = ∫abf(x)dx[/tex]

Since the given function is y = 2x-3, we can write it as y = 2x^(-3) by applying the power rule.

Hence,A = [tex]∫62x^(-3)dx = [-2x^(-2)]6t = -2/t^2 + 2/36[/tex]We need to evaluate this area for t = 10 and t = 100, so we get[tex]A = -2/10^2 + 2/36 = -1/25 + 1/18 = 7/450andA = -2/100^2 + 2/36 = -1/5000 + 1/18 = 449/4500[/tex]Total area under this curve for x ≥ 6

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A box with a square base and open top must have a volume of 62500 cm3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible. A(x) = Next, find the derivative, A'(x). A'(x) = Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x² .] A'(x) = 0 when x =

Answers

The area of the square base = x².

we have:l = w = x ... (2) ... And, h = V/lw = V/x² ... (3) ...

The dimension of the box that minimizes the amount of material used is x =  (2V)1/3. A(x) = x² + 4V/x, A'(x) = 2x - 4V/x², x =  (2V)1/3

The given volume of the box is 62500 cm³. We wish to find the dimensions of the box that minimize the amount of material used.

To obtain the formula for the surface area of the box in terms of only x, the length of one side of the square base, we use the formula for the volume of a box:V = lwh ... (1) ... where V is the volume, l is the length, w is the width, and h is the height of the box. Here, the base of the box is a square with side length x.

Hence, the area of the square base = x². Therefore, we have:l = w = x ... (2) ... And, h = V/lw = V/x² ... (3) ... We can substitute (2) and (3) in (1) to get the formula for V in terms of x as follows:V = x² V/x² A(x) = A(x) = x² + 4xhA(x) = x² + 4x(V/x²) = x² + 4V/x

Now, to find the derivative A'(x) of A(x), we differentiate A(x) with respect to x:A'(x) = 2x - 4V/x²  A'(x) = 0 when x =  (2V)1/3. Therefore, the dimension of the box that minimizes the amount of material used is x =  (2V)1/3. A(x) = x² + 4V/x, A'(x) = 2x - 4V/x², x =  (2V)1/3

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