Tüm alıştırma soruları

Soru 4901Soru →

Consider the exponential expression $(y^7)^3$ . If this expression is rewritten in the equivalent form $y^n$ , what is the numerical value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 21; twenty-one; n=21; n = 21

Cevap

The numerical value of the exponent is 21.

When raising a power to another power, the laws of exponents dictate that the two exponents must be multiplied. Multiplying the inside exponent 7 by the outside exponent 3 yields 21.

Adım Adım Çözüm

1

Identify the appropriate exponent rule to simplify the expression

(y^7)^3

.

The expression requires the Power of a Power property.

An exponential term with a base and an exponent is being raised entirely to another power.

2

Apply the Power of a Power rule, which states that

(x^a)^b = x^{a \times b}

.

Multiply the inner exponent by the outer exponent:

7 \times 3

.

To raise a power to another power, you must multiply the exponents together.

3

Calculate the final numerical value.

7 \times 3 = 21

, meaning the expression simplifies to

y^{21}

.

By matching

y^{21}

to the given form

y^n

, we determine that

n = 21

.

Anahtar Kavram

Power of a Power Rule for Exponents

Tahmini Süre:30s

Soru 4902Soru →

Evaluate the product of the monomial expressions $4z^5$ and $2z^3$ . Which of the following represents the simplified result?

8z^8

8z^{15}

6z^8

6z^{15}

Cevabı ve açıklamayı göster

Cevap: $8z^8$

Cevap

The expression simplifies to

8z^8

.

When multiplying two monomial expressions, the numerical coefficients are multiplied together (

4 \times 2 = 8

), and the exponents of the common variable base are added together according to the product rule for exponents (

5 + 3 = 8

). This results in

8z^8

.

Adım Adım Çözüm

1

Identify the operation to be performed between the two terms.

The terms

4z^5

and

2z^3

are being multiplied.

Understanding the operation is required to apply the correct mathematical rules for coefficients and exponents.

2

Multiply the numerical coefficients.

4 \times 2 = 8

Coefficients in a product of monomials are multiplied together normally.

3

Multiply the variable parts by applying the product rule for exponents.

z^5 \times z^3 = z^{5+3} = z^8

When multiplying exponential expressions with the same base, their exponents must be added.

4

Combine the calculated coefficient and the variable part.

8z^8

This brings the simplified pieces together into the final term.

Anahtar Kavram

Operations with Variable Exponents

Soru 4903Soru →

An aviation squadron requires $3$ hours of preventive maintenance for every $8$ hours of flight time. If a specific helicopter undergoes $24$ hours of preventive maintenance during a given month, what is the total number of hours logged for both flight and maintenance combined?

33

64

88

192

Cevabı ve açıklamayı göster

Cevap: $88$

Cevap

The total number of hours logged is

88

.

The given ratio of maintenance to flight time is

3:8

. Because the helicopter had

24

hours of maintenance, we find the multiplier by dividing

24

by

3

, which is

8

. We then apply this multiplier to the flight time ratio (

8 \times 8 = 64

hours). To find the total combined hours, we add the maintenance hours and flight hours together:

24 + 64 = 88

.

Adım Adım Çözüm

1

Find the scale factor (multiplier) by comparing the actual maintenance hours to the ratio.

24 \div 3 = 8

This establishes how many times larger the actual numbers are compared to the base ratio.

2

Calculate the actual flight hours using the scale factor.

8 \times 8 = 64

hours of flight time

Applying the scale factor to the flight time ratio gives the actual flight time.

3

Add the actual maintenance hours and actual flight hours to find the total time.

24 + 64 = 88

total hours

The question specifically asks for the combined total of both flight and maintenance time.

Anahtar Kavram

Solving part-to-part ratio word problems and calculating a total sum.

Soru 4904Soru →

Consider the polynomial expression provided below:

(2x - 3)(x + 4) - 2(x^2 - 5x + 1)

Which of the following represents its fully simplified equivalent?

10x - 14

-5x - 14

15x - 14

x

Cevabı ve açıklamayı göster

Cevap: $15x - 14$

Cevap

The completely simplified equivalent expression is

15x - 14

.

The expression correctly expands to

2x^2 + 5x - 12

on the left and

-2x^2 + 10x - 2

on the right. When adding these together, the

x^2

terms cancel out, the

x

terms sum to

15x

, and the constants sum to

-14

, leaving

15x - 14

.

Adım Adım Çözüm

1

Expand

(2x - 3)(x + 4)

using the distributive property.

2x^2 + 8x - 3x - 12

, which simplifies to

2x^2 + 5x - 12

.

To eliminate the first set of parentheses by multiplying each term in the first binomial by each term in the second binomial.

2

Distribute

-2

to each term inside the trinomial

(x^2 - 5x + 1)

.

-2x^2 + 10x - 2

.

To eliminate the second set of parentheses, carefully applying the negative sign to all inner terms.

3

Combine the expanded expressions.

(2x^2 + 5x - 12) + (-2x^2 + 10x - 2)

.

To prepare the entire expression for the collection of like terms.

4

Group and combine like terms (

x^2

terms,

x

terms, and constants).

(2x^2 - 2x^2) + (5x + 10x) + (-12 - 2) = 0x^2 + 15x - 14 = 15x - 14

.

To achieve the final simplified standard form.

Anahtar Kavram

Expanding polynomials, distributing negative coefficients, and combining like terms.

Soru 4905Soru →

Complete the algebraic equation that accurately translates the given statement.

Aşağıdaki boşlukları doldurun

Statement: 'Five subtracted from four times a number

k

equals eleven.'

Equation: - = 11

Cevabı ve açıklamayı göster

Cevap

The first blank is '4k' and the second blank is '5', forming the correct equation 4k - 5 = 11.

The phrase 'four times a number

k

' becomes

4k

. The phrase 'Five subtracted from' means 5 is taken away from

4k

, so the 5 must be placed after the minus sign, forming

4k - 5

. Finally, 'equals eleven' sets the expression equal to 11. Therefore, to match the provided equation structure of '___ - ___ = 11', the first blank must be filled with '4k' and the second blank must be filled with '5'.

Adım Adım Çözüm

1

Translate the multiplication part of the statement.

'four times a number

k

' translates to the algebraic expression

4k

.

The word 'times' indicates multiplication between the coefficient 4 and the variable

k

.

2

Apply the subtraction part of the statement.

'Five subtracted from' means 5 is taken away from the previous term, resulting in

4k - 5

.

The phrase 'subtracted from' is a turnaround phrase, which means the 5 must be placed after the minus sign, reversing the spoken order.

3

Complete the equation.

Setting the expression equal to 11 gives the full equation:

4k - 5 = 11

.

The word 'equals' indicates where the equal sign (=) should be placed in relation to the constant 11.

Anahtar Kavram

Translating turnaround words (like 'subtracted from' or 'less than') and basic multiplication phrases into an algebraic equation.

Tahmini Süre:30s

Soru 4906Soru →

Which of the following describes all values of $x$ that satisfy the inequality below?

2(x - 4) - 5x < x + 4

x > -3

x < -3

x > 3

x > 1

Cevabı ve açıklamayı göster

Cevap: $x > -3$

Cevap

The correct solution is

x > -3

.

The correct answer accurately simplifies the equation down to

-4x < 12

. Because isolating the variable in the final step requires dividing by

-4

, the less than symbol (

<

) must be reversed to a greater than symbol (

>

), resulting in

x > -3

.

Adım Adım Çözüm

1

Distribute the 2 into the parentheses on the left side.

2x - 8 - 5x < x + 4

To clear the parentheses and access the terms inside.

2

Combine the like terms (

2x

and

-5x

) on the left side.

-3x - 8 < x + 4

To simplify the left side of the inequality.

3

Subtract

x

from both sides to group the variable terms.

-4x - 8 < 4

To collect all terms containing

x

on one side.

4

Add 8 to both sides of the inequality.

-4x < 12

To isolate the term with the variable.

5

Divide both sides by -4 and reverse the inequality sign.

x > -3

When multiplying or dividing an inequality by a negative number, the inequality symbol must be flipped to maintain a mathematically true statement.

Anahtar Kavram

Solving linear inequalities with variables on both sides requires combining like terms and reversing the inequality symbol when dividing by a negative coefficient.

Soru 4907Soru →

Match each unsimplified algebraic expression on the left with its completely simplified equivalent on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correctly simplified expressions are:

5x - 3(2x - 4)

matches with

-x + 12

;

3(2m - 5n) - 2(m - 3n)

matches with

4m - 9n

;

4x(x - 2) - x^2 + 5x

matches with

3x^2 - 3x

; and

7y - 3[y - 2(y + 1)]

matches with

10y + 6

.

Each unsimplified expression must be correctly expanded using the distributive property, ensuring that negative multipliers are applied to every term inside the parentheses. After expanding, grouping and adding the coefficients of like terms produces the single correct simplified equivalent provided in the match list.

Adım Adım Çözüm

1

Distribute any coefficients or negative signs immediately outside of parentheses or brackets.

Terms are expanded (e.g.,

-3(2x - 4)

becomes

-6x + 12

).

Distribution removes grouping symbols, which is required before combining terms.

2

For expressions with nested grouping symbols, simplify from the innermost parentheses outward.

Complex expressions like

7y - 3[y - 2(y + 1)]

simplify step-by-step to

7y + 3y + 6

.

Following the order of operations ensures accurate distribution of multipliers.

3

Group and combine like terms (terms with the exact same variable and exponent).

The final canonical form is reached for each expression.

Combining like terms fully simplifies the polynomial into standard form.

Anahtar Kavram

Simplifying Algebraic Expressions

Tahmini Süre:1m 30s

Soru 4908Soru →

A military supply shipment contains $3.4 \times 10^4$ ready-to-eat meals. What is the total number of meals written in standard numerical form?

Cevabı ve açıklamayı göster

Cevap: 34000

Cevap

34000

The expression

3.4 \times 10^4

represents

3.4

multiplied by

10,000

. Shifting the decimal point 4 places to the right results in exactly

34,000

.

Adım Adım Çözüm

1

Identify the exponent on the base 10.

The exponent is 4.

The exponent determines the number of places the decimal point must be shifted.

2

Move the decimal point 4 places to the right.

3.4 becomes 34000.

Multiplying by 10^4 is equivalent to shifting the decimal to the right by 4 places to convert the scientific notation into standard form.

Anahtar Kavram

Converting from scientific notation to standard form involves shifting the decimal point based on the value and sign of the exponent.

Soru 4909Soru →

A linear equation is structured as follows:

\frac{1}{2}(4y - 6) - \frac{2}{3}(9 - 6y) = 2y + 15

Based on this relationship, what is the correct value of $y$ ?

-6

-3

3

6

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct value is found by properly distributing the fractional coefficients, carefully tracking negative signs, and preserving equality to isolate the variable. The operations simplify the initial equation to 6y - 9 = 2y + 15, which reduces to 4y = 24, resulting in 6.

Adım Adım Çözüm

1

Distribute

\frac{1}{2}

into the first set of parentheses.

2y - 3

Applying the distributive property eliminates the first group.

2

Distribute

-\frac{2}{3}

into the second set of parentheses.

-6 + 4y

Multiplying the negative fraction across the terms. Notice that multiplying two negative values (

-2/3

and

-6y

) yields a positive result (

+4y

).

3

Combine the distributed terms on the left side of the equation.

(2y + 4y) + (-3 - 6) = 6y - 9

Combining like terms simplifies the left side of the equation before attempting to isolate the variable.

4

Set the simplified left side equal to the right side and solve for y.

6y - 9 = 2y + 15 \Rightarrow 4y - 9 = 15 \Rightarrow 4y = 24 \Rightarrow y = 6

Using inverse operations groups the variable terms on one side and constant terms on the other to reveal the final value.

Anahtar Kavram

Solving complex linear equations involving fractional distribution and sign management.

Soru 4910Soru →

Calculate the actual length of the property line based on the given scale and enter the exact numerical value.

Aşağıdaki boşlukları doldurun

A city surveyor is examining a property deed map that uses a scale of

1\frac{1}{2}\text{ inches} = 40\text{ feet}

. If the eastern boundary line measures

5\frac{1}{4}\text{ inches}

on this map, the actual length of the boundary line is feet.

Cevabı ve açıklamayı göster

Cevap

140

The correct answer is found by setting up a direct proportion based on the scale:

\frac{1.5\text{ inches}}{40\text{ feet}} = \frac{5.25\text{ inches}}{x\text{ feet}}

. Cross-multiplying gives

1.5x = 210

, and dividing by

1.5

gives an actual length of

140

feet.

Adım Adım Çözüm

1

Convert the mixed fractions into decimals for easier algebraic calculation.

The map scale is

1.5\text{ inches} = 40\text{ feet}

. The measured boundary is

5.25\text{ inches}

.

Working with decimals simplifies the process of setting up and solving a proportion.

2

Set up a direct proportion comparing the map distance to the actual distance.

\frac{1.5\text{ inches}}{40\text{ feet}} = \frac{5.25\text{ inches}}{x\text{ feet}}

A map's scale establishes a consistent ratio between the dimensions on the drawing and the true physical dimensions.

3

Cross-multiply the terms to isolate the unknown variable

x

.

1.5x = 40 \times 5.25

Cross-multiplication transforms the proportion into a linear equation that is easy to solve.

4

Multiply the values on the right side of the equation.

40 \times 5.25 = 210

Simplifying the right side to

210

leaves a basic one-step algebraic equation:

1.5x = 210

.

5

Divide both sides by

1.5

to solve for

x

.

x = \frac{210}{1.5} = 140

Dividing by the coefficient isolates

x

, revealing the actual length of the boundary line in feet.

Anahtar Kavram

Using proportional reasoning and cross-multiplication to convert map measurements to real-world distances.

Soru 4911Soru →

What is the completely simplified form of the algebraic expression $7x - 4(2x - 3y) + 5y$ ?

-x - 7y

-x + 2y

-x + 17y

7x + 4xy + 5y

Cevabı ve açıklamayı göster

Cevap: $-x + 17y$

Cevap

-x + 17y

To completely simplify the expression, you must first distribute the

-4

across the terms in the parentheses. Multiplying

-4

by

2x

gives

-8x

, and multiplying

-4

by

-3y

gives

+12y

(since a negative times a negative is a positive). Substituting this back gives

7x - 8x + 12y + 5y

. Finally, group the like variables:

7x - 8x = -x

, and

12y + 5y = 17y

. Combining these gives

-x + 17y

.

Adım Adım Çözüm

1

Distribute the

-4

to both terms inside the parentheses

(2x - 3y)

.

-8x + 12y

The distributive property must be applied first to eliminate the parentheses, paying careful attention to the multiplication of two negative numbers (

-4

and

-3y

).

2

Rewrite the full expression incorporating the expanded terms.

7x - 8x + 12y + 5y

This sets up the expression so that all individual terms are clearly visible for grouping.

3

Combine the like terms for both

x

and

y

.

-x + 17y

Adding the coefficients of identical variables (

7x - 8x = -x

and

12y + 5y = 17y

) provides the expression in its final standard form.

Anahtar Kavram

Simplifying Algebraic Expressions via Distributive Property and Like Terms

Soru 4912Soru →

Pair each basic algebraic expression with its correctly simplified result by applying the fundamental laws of exponents.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings are:

m^6 \cdot m^2

with

m^8

;

\frac{n^8}{n^5}

with

n^3

;

(p^4)^3

with

p^{12}

; and

q^{-5}

with

\frac{1}{q^5}

.

Each expression can be simplified by directly applying one specific fundamental property of exponents.

m^6 \cdot m^2

uses the product rule.

\frac{n^8}{n^5}

uses the quotient rule.

(p^4)^3

uses the power rule.

q^{-5}

uses the negative exponent definition.

Adım Adım Çözüm

1

Simplify the product expression

m^6 \cdot m^2

.

m^8

When bases are identical and multiplied, their exponents are added together.

2

Simplify the quotient expression

\frac{n^8}{n^5}

.

n^3

When dividing identical bases, the exponent of the denominator is subtracted from the exponent of the numerator.

3

Simplify the nested power expression

(p^4)^3

.

p^{12}

When raising an exponent to another power, the two exponents are multiplied.

4

Rewrite the negative exponent expression

q^{-5}

.

\frac{1}{q^5}

A negative exponent indicates taking the reciprocal of the base raised to the positive exponent.

Anahtar Kavram

Applying fundamental exponent rules (product, quotient, power, and negative exponents) to simplify single-variable expressions.

Soru 4913Soru →

A naval shipyard allocates its weekly steel supply for submarine repair, surface ship repair, and new construction in a ratio of $3:4:5$ . If the surface ship repair division receives $240$ tons of steel, what is the total amount of steel allocated to all three divisions combined?

480

576

720

960

Cevabı ve açıklamayı göster

Cevap: 720

Cevap

The total amount of steel allocated to all three divisions is 720 tons.

The ratio of steel for submarine repair, surface ship repair, and new construction is

3:4:5

, which means there are

12

total parts (

3+4+5

). We are given that surface ship repair, the middle term representing

4

parts of the ratio, receives

240

tons. To find the value of a single part, divide

240

by

4

, yielding

60

tons. Finally, multiply this unit rate by the

12

total parts to find the total steel allocation:

60 \times 12 = 720

tons.

Adım Adım Çözüm

1

Identify the number of ratio parts assigned to surface ship repair and calculate the total number of parts.

Surface ship repair = 4 parts. Total parts = 3 + 4 + 5 = 12 parts.

To find the value of a single part, we must correctly match the given amount to its corresponding ratio value.

2

Calculate the amount of steel that corresponds to one ratio part.

240 tons / 4 parts = 60 tons per part.

Finding this unit rate allows us to scale up to the total amount.

3

Multiply the unit rate by the total number of parts.

60 tons/part * 12 parts = 720 tons.

The question asks for the total amount of steel allocated to all three divisions combined.

Anahtar Kavram

Solving multi-part ratio problems by finding a unit rate

Alternatif Yöntem

You can set up a direct proportion: 4 parts / 240 tons = 12 total parts / x tons. Cross-multiplying gives 4x = 2880, so dividing both sides by 4 yields x = 720.

Tahmini Süre:1m 0s

Soru 4914Soru →

A mechanical engineer is determining the load factor for a support beam. The calculation requires evaluating the equation below.

\text{Load Factor} = \left( 5 \frac{5}{8} - 2 \frac{3}{4} \right) \div \left( 1 \frac{1}{3} \times \frac{3}{8} \right)

What is the final load factor? Express your answer as an exact decimal.

Cevabı ve açıklamayı göster

Cevap: 5.75

Cevap

The final load factor is exactly 5.75.

The expression evaluates to exactly 5.75. This is found by first converting all mixed numbers to improper fractions, calculating the values inside the parentheses (

\frac{23}{8}

and

\frac{1}{2}

respectively), and then performing fraction division by multiplying by the reciprocal (

\frac{23}{8} \times 2 = \frac{23}{4} = 5.75

).

Adım Adım Çözüm

1

Convert the mixed numbers in the expression to improper fractions.

5 \frac{5}{8} = \frac{45}{8}

,

2 \frac{3}{4} = \frac{11}{4}

, and

1 \frac{1}{3} = \frac{4}{3}

.

Fraction multiplication and division require working with improper fractions rather than mixed numbers.

2

Perform the subtraction inside the first set of parentheses:

\frac{45}{8} - \frac{11}{4}

.

Convert

\frac{11}{4}

to

\frac{22}{8}

. Then,

\frac{45}{8} - \frac{22}{8} = \frac{23}{8}

.

To subtract fractions, they must have a common denominator. The lowest common denominator for 8 and 4 is 8.

3

Perform the multiplication inside the second set of parentheses:

\frac{4}{3} \times \frac{3}{8}

.

Multiply the numerators and denominators:

\frac{4 \times 3}{3 \times 8} = \frac{12}{24}

. This simplifies to

\frac{1}{2}

.

Following the order of operations (PEMDAS), operations inside parentheses must be evaluated first.

4

Divide the result of the first parentheses by the result of the second parentheses.

\frac{23}{8} \div \frac{1}{2} = \frac{23}{8} \times \frac{2}{1} = \frac{46}{8}

.

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

5

Simplify the final fraction and convert it to a decimal format.

\frac{46}{8} = \frac{23}{4} = 5 \frac{3}{4}

. As a decimal,

5 \frac{3}{4} = 5.75

.

The question specifically requests the final load factor expressed as a decimal.

Anahtar Kavram

Compound Fraction Operations and Order of Operations

Soru 4915Soru →

Which value of $m$ is the solution to the equation below?

6m - 12 = 2m + 16

7

1

-2

-7

Cevabı ve açıklamayı göster

Cevap: $7$

Cevap

The correct solution is

7

.

By subtracting

2m

from both sides and then adding 12 to both sides, the equation simplifies to

4m = 28

. Dividing by 4 gives the correct result of

7

.

Adım Adım Çözüm

1

Subtract

2m

from both sides of the equation to group the variable terms on the left side.

4m - 12 = 16

This keeps all instances of the unknown variable

m

on one side.

2

Add 12 to both sides of the equation to isolate the variable term.

4m = 28

This cancels out the constant on the left side, leaving only the term with

m

.

3

Divide both sides by 4 to solve for

m

.

m = 7

This removes the coefficient, revealing the final value of the variable.

Anahtar Kavram

Solving One-Variable Linear Equations

Tahmini Süre:45s

Soru 4916Soru →

Calculate the value of the unknown variable.

Aşağıdaki boşlukları doldurun

Calculate the value of

k

that makes the following equation true:

\frac{3}{4}(8k - 12) - \frac{2}{5}(15 - 5k) = 3(k + 2) + 4

k =

Cevabı ve açıklamayı göster

Cevap

5

The correct value is 5. By distributing the fractions on the left, the expression becomes

6k - 9 - 6 + 2k

. Combining like terms simplifies this to

8k - 15

. On the right side, distributing the 3 results in

3k + 6 + 4

, which simplifies to

3k + 10

. Equating both simplified sides gives

8k - 15 = 3k + 10

. Subtracting

3k

from both sides leaves

5k - 15 = 10

, and adding

15

to both sides yields

5k = 25

. Dividing by 5 results in

k = 5

.

Adım Adım Çözüm

1

Distribute the fractional terms on the left side of the equation.

6k - 9 - 6 + 2k

Multiplying

\frac{3}{4}

by

(8k - 12)

yields

6k - 9

. Multiplying

-\frac{2}{5}

by

(15 - 5k)

yields

-6 + 2k

. Paying close attention to the negative sign during distribution is critical.

2

Distribute the terms on the right side of the equation.

3k + 6 + 4

Multiplying

3

by

(k + 2)

expands the expression, setting up the right side for simplification.

3

Combine like terms on both sides of the equal sign.

8k - 15 = 3k + 10

On the left side, combine the variable terms (

6k + 2k = 8k

) and constant terms (

-9 - 6 = -15

). On the right side, combine the constant terms (

6 + 4 = 10

).

4

Isolate the variable

k

to find its value.

5k = 25

, which simplifies to

k = 5

.

Subtract

3k

from both sides to group the variables on the left (

5k - 15 = 10

). Then, add

15

to both sides to group the constants on the right (

5k = 25

). Dividing both sides by

5

yields the final solution.

Anahtar Kavram

Solving One-Variable Linear Equations

Soru 4917Soru →

A commercial facility is securing a right-triangular storage lot. The two perpendicular boundaries of the lot measure $60$ yards and $80$ yards. Security fencing will be installed along the entire perimeter to create a closed loop. If a support post must be placed every $12$ feet, how many total posts are required to complete the fence?

20

35

60

200

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

The correct answer is found by first determining the hypotenuse of the right-triangular lot, which is 100 yards (

60^2 + 80^2 = 100^2

). The total perimeter is 240 yards (60 + 80 + 100). Because the post spacing is given in feet, the perimeter must be converted to 720 feet (240 × 3). Finally, dividing the 720-foot boundary by the 12-foot spacing requirement yields exactly 60 posts.

Adım Adım Çözüm

1

Find the length of the third side of the lot using the Pythagorean theorem (

a^2 + b^2 = c^2

).

60^2 + 80^2 = 3600 + 6400 = 10000

. The square root of

10000

is

100

yards.

The boundary of a right-triangular lot includes the hypotenuse, which must be found to calculate the total perimeter.

2

Calculate the total perimeter of the triangular lot.

60 + 80 + 100 = 240

yards.

The fencing will enclose the entire boundary of the lot.

3

Convert the perimeter into feet to match the unit of the post spacing.

240 \text{ yards} \times 3 \text{ feet/yard} = 720 \text{ feet}

.

Measurements must be in the same unit (feet) before dividing.

4

Divide the total perimeter by the distance required between each post.

720 \div 12 = 60

posts.

For a closed loop, the total number of posts equals the total perimeter divided by the spacing distance.

Anahtar Kavram

Calculating the perimeter of a right triangle using the Pythagorean theorem, converting units of measurement, and determining spacing requirements for closed boundaries.

Alternatif Yöntem

Instead of converting the perimeter to feet, convert the post spacing to yards. Since 3 feet equals 1 yard, 12 feet equals 4 yards. Divide the 240-yard perimeter by 4 yards to get 60 posts.

Tahmini Süre:1m 30s

Soru 4918Soru →

Arrange the following values from least to greatest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence from least to greatest is:

620

,

8,500

,

1.2 \times 10^4

,

3.0 \times 10^5

.

By converting all values to standard form, it is clear that

620

is the smallest, followed by

8,500

, then

12,000

(which is

1.2 \times 10^4

), and finally

300,000

(which is

3.0 \times 10^5

).

Adım Adım Çözüm

1

Convert all values to standard numerical form to easily compare their sizes.

The values become:

12,000

(from item 1),

620

(from item 2),

300,000

(from item 3), and

8,500

(from item 4).

Translating scientific notation into standard form allows for direct visual comparison of digit lengths and magnitudes.

2

Group and compare the numbers by their total number of digits.

620

has three digits,

8,500

has four digits,

12,000

has five digits, and

300,000

has six digits.

For whole positive numbers, a value with fewer digits is always smaller than a value with more digits.

3

Order the values sequentially starting from the smallest.

The final order is

620

<

8,500

<

12,000

<

300,000

.

This establishes the ascending sequence requested by the question.

Anahtar Kavram

Comparing Magnitudes in Standard and Scientific Notation

Alternatif Yöntem

Instead of using standard form, you can convert all numbers to scientific notation:

6.2 \times 10^2

,

8.5 \times 10^3

,

1.2 \times 10^4

,

3.0 \times 10^5

. Because the exponents (

2, 3, 4, 5

) are already in ascending order, you can immediately determine the correct sequence without looking at the coefficients.

Tahmini Süre:45s

Soru 4919Soru →

A cybersecurity team analyzes network traffic and categorizes detected threats into low, medium, and high severity in a ratio of $9:5:2$ . If the team logged exactly $145$ medium severity threats during a shift, what was the total number of threats logged across all three categories?

Cevabı ve açıklamayı göster

Cevap: 464

Cevap

464

The correct answer is found by first determining the value of a single ratio part. Since

5

parts represent

145

medium threats, one part equals

29

threats. Multiplying this base unit by the total number of parts (

16

) gives the total of

464

threats.

Adım Adım Çözüm

1

Determine the total number of parts in the ratio.

9 + 5 + 2 = 16

parts

To find the total amount of threats, we first need to know how many total parts make up the complete ratio.

2

Find the numerical value of a single part.

145 \div 5 = 29

threats per part

We know that

5

parts equal

145

medium threats. Dividing the count by the parts gives the baseline value of one part.

3

Calculate the total number of threats.

29 \times 16 = 464

threats

Multiplying the value of one part by the total number of parts yields the total count across all categories.

Anahtar Kavram

Solving Part-to-Whole Ratio Problems

Soru 4920Soru →

Determine the numerical value of the variable that satisfies the given linear equation.

Aşağıdaki boşlukları doldurun

For the equation

7k - 5 = 3k + 19

, the value of

k

is .

Cevabı ve açıklamayı göster

Cevap

The correct value for k is 6.

To find the value of

k

, we must isolate the variable on one side of the equation. By subtracting

3k

from both sides, we get

4k - 5 = 19

. Next, adding

5

to both sides gives

4k = 24

. Finally, dividing both sides by

4

results in

k = 6

.

Adım Adım Çözüm

1

Subtract

3k

from both sides of the equation.

4k - 5 = 19

This groups all variable terms on the left side of the equation.

2

Add

5

to both sides of the equation.

4k = 24

This isolates the variable term by moving the constant to the right side.

3

Divide both sides by

4

.

k = 6

This cancels the coefficient on the variable, yielding the final solution.

Anahtar Kavram

Solving linear equations with variables on both sides by using inverse operations to isolate the unknown.

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