Tüm alıştırma soruları

Soru 1481Soru →

A moving company uses the equation $C = 2.5d + 75$ to determine the total charge $C$ , in dollars, for renting a small truck and driving it $d$ miles. According to the relationship, what is the charge, in dollars, for each mile the truck is driven?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The charge for each mile the truck is driven is 2.5 dollars.

In the linear model

C = 2.5d + 75

, the total cost

C

is a function of the number of miles driven

d

. The rate of change of this function represents the cost per mile driven. In the equation, this rate of change is the coefficient of

d

, which is 2.5. Therefore, the charge for each mile the truck is driven is 2.5 dollars.

Adım Adım Çözüm

1

Analyze the linear equation

C = 2.5d + 75

to determine the relationship between variables.

The total charge

C

depends on the number of miles

d

driven, with a rate of change of 2.5 dollars per mile and a base fee of 75 dollars.

Understanding the components of a linear equation helps isolate the rate of change.

2

Identify the coefficient of the independent variable

d

.

The coefficient of

d

is 2.5.

In a linear equation of the form

y = mx + b

, the coefficient of the independent variable represents the rate of change.

3

Interpret the meaning of this coefficient in the context of the problem.

The coefficient 2.5 represents the cost, in dollars, incurred per mile driven.

The question asks for the charge per mile, which corresponds to the rate of change.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 1482Soru →

A system of two equations is given:

y = x^2 + kx + 4

y = 4x - k

In these equations, $k$ represents a positive constant. If the system has a single real solution, what is the value of $k$ ?

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Cevap: 12

Cevap

12

Setting the two equations equal yields the quadratic equation

x^2 + (k - 4)x + (k + 4) = 0

. For the system to have exactly one real solution, the discriminant of this quadratic equation must be equal to

0

. The discriminant is

(k - 4)^2 - 4(1)(k + 4) = k^2 - 12k

. Solving

k^2 - 12k = 0

yields

k = 0

or

k = 12

. Since

k

must be positive, the value of

k

is

12

.

Adım Adım Çözüm

1

Set the two equations equal to each other to form a single quadratic equation in terms of

x

.

x^2 + (k - 4)x + (k + 4) = 0

Equating the expressions for

y

allows us to find the

x

-coordinates where the graphs of the two equations intersect.

2

Set the discriminant of the quadratic equation to zero.

(k - 4)^2 - 4(1)(k + 4) = 0

A system of equations consisting of a line and a parabola has a single real solution if and only if the line is tangent to the parabola, which corresponds to a quadratic equation with a discriminant of zero.

3

Expand and simplify the equation for

k

.

k^2 - 12k = 0

Expanding

(k - 4)^2

yields

k^2 - 8k + 16

, and distributing

-4

yields

-4k - 16

. Combining like terms simplifies the relation.

4

Solve for

k

and apply the constraint that

k > 0

.

k = 12

Factoring

k(k - 12) = 0

gives

k = 0

or

k = 12

. Since

k

is specified to be positive,

k = 12

is the correct value.

Anahtar Kavram

Solving nonlinear systems of equations by substitution and using the discriminant to find conditions for a single real solution.

Soru 1483Soru →

A software programmer is writing test cases for a new application. The programmer must write at least 15 test cases in total, consisting of $x$ unit tests and $y$ integration tests. Each unit test takes 10 minutes to write, and each integration test takes 30 minutes to write. If the programmer has at most 300 minutes to write all the test cases, what is the maximum number of integration tests the programmer can write?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Adım Adım Çözüm

1

Set up the system of inequalities representing the given constraints.

x + y \ge 15

and

10x + 30y \le 300

The total number of tests must be at least 15, and the total time taken by writing

x

unit tests (10 minutes each) and

y

integration tests (30 minutes each) cannot exceed 300 minutes.

2

Simplify the time inequality and combine it with the total test count constraint to isolate

y

.

x + 3y \le 30

. Substituting

x \ge 15 - y

into this inequality yields

(15 - y) + 3y \le 30 \implies 15 + 2y \le 30

.

Simplification and substitution help find the upper bound for the number of integration tests.

3

Solve for

y

and determine the maximum integer value.

2y \le 15 \implies y \le 7.5

. The largest integer satisfying this inequality is 7.

The number of integration tests must be a whole number, so we round down to the nearest integer.

4

Verify that a valid integer number of unit tests (

x

) exists when

y = 7

.

When

y = 7

, we get

x \ge 15 - 7 \implies x \ge 8

and

x + 3(7) \le 30 \implies x \le 9

. The integers

x = 8

and

x = 9

both satisfy the conditions.

We must confirm that the maximum value of

y

is achievable with an integer number of unit tests.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 1484Soru →

If $(x, y)$ is a solution to the system of equations below and $x > 0$ , what is the value of $x + y$ ?

\begin{aligned} x^2 + y^2 &= 50 \\ x - y &= 6 \end{aligned}

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Solving the linear equation for

x

gives

x = y + 6

. Substituting this into the circle equation yields

(y + 6)^2 + y^2 = 50

, which simplifies to

2y^2 + 12y - 14 = 0

. Dividing by

2

gives

y^2 + 6y - 7 = 0

, which factors as

(y + 7)(y - 1) = 0

. This yields

y = 1

(with

x = 7

) or

y = -7

(with

x = -1

). The constraint

x > 0

requires selecting the solution

(7, 1)

. The sum of

x

and

y

for this solution is

7 + 1 = 8

.

Adım Adım Çözüm

1

Express

x

in terms of

y

using the linear equation.

x = y + 6

This allows for substitution into the second equation to reduce the system to a single variable.

2

Substitute the expression for

x

into the equation of the circle and expand.

2y^2 + 12y - 14 = 0

Substituting

x = y + 6

into

x^2 + y^2 = 50

yields

(y + 6)^2 + y^2 = 50

. Expanding gives

y^2 + 12y + 36 + y^2 = 50

.

3

Simplify and solve the quadratic equation by factoring.

y = 1

or

y = -7

Dividing the equation by 2 gives

y^2 + 6y - 7 = 0

, which factors as

(y + 7)(y - 1) = 0

.

4

Calculate the corresponding values of

x

for each

y

.

For

y = 1

,

x = 7

; for

y = -7

,

x = -1

Using the relation

x = y + 6

yields the coordinates of the two intersection points.

5

Apply the constraint

x > 0

and calculate

x + y

.

x + y = 8

Since

x > 0

, the only valid solution is

(7, 1)

. The sum of the coordinates is

7 + 1 = 8

.

Anahtar Kavram

Solving a nonlinear system of equations representing a circle and a line using substitution and factoring.

Soru 1485Soru →

In the $xy$ -plane, the graph of the linear equation $3x + 5y = c$ , where $c$ is a constant, passes through the point $(4, 3)$ . What is the $x$ -coordinate of the point on this graph where the $y$ -coordinate is $-3$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The correct answer is 14.

To find the correct answer, first substitute the given point

(4, 3)

into the equation

3x + 5y = c

to determine the value of the constant

c

:

3(4) + 5(3) = 12 + 15 = 27

. Thus, the equation is

3x + 5y = 27

. Next, substitute

-3

for

y

in this equation to find the corresponding

x

-coordinate:

3x + 5(-3) = 27 \implies 3x - 15 = 27

. Adding

15

to both sides gives

3x = 42

, and dividing by

3

yields

x = 14

.

Adım Adım Çözüm

1

Substitute the point

(4, 3)

into the equation

3x + 5y = c

to solve for

c

.

c = 27

Since the point lies on the graph of the equation, its coordinates must satisfy the equation.

2

Substitute

y = -3

and

c = 27

into the equation

3x + 5y = c

.

3x - 15 = 27

We want to find the

x

-coordinate of the point on the line when the

y

-coordinate is

-3

.

3

Solve the equation

3x - 15 = 27

for

x

.

x = 14

Isolating

x

gives the

x

-coordinate of the point.

Anahtar Kavram

Using a known point on a line to find a constant coefficient or constant term, and using the resulting equation to find other coordinates.

Alternatif Yöntem

Alternatively, you can write the equation in slope-intercept form. Solving

3x + 5y = c

for

y

gives

y = -\frac{3}{5}x + \frac{c}{5}

. The slope of the line is

-\frac{3}{5}

. Since the slope is constant, the change in

y

divided by the change in

x

between

(4, 3)

and

(x, -3)

is equal to the slope:

\frac{-3 - 3}{x - 4} = -\frac{3}{5} \implies \frac{-6}{x - 4} = -\frac{3}{5}

. Cross-multiplying gives

-30 = -3(x - 4) \implies 10 = x - 4 \implies x = 14

.

Tahmini Süre:1m 30s

Soru 1486Soru →

For all $x > 3$ , which of the following expressions is equivalent to $\frac{x^2 - 9}{x^2 - 2x - 3} \cdot \frac{x + 1}{x}$ ?

\frac{x+3}{x+1}

\frac{x+3}{x}

\frac{x-3}{x}

\frac{x^2+4x+3}{x^2-x}

Cevabı ve açıklamayı göster

Cevap: $\frac{x+3}{x}$

Cevap

\frac{x+3}{x}

Factoring the numerator of the first fraction as

(x-3)(x+3)

and its denominator as

(x-3)(x+1)

allows the common factor

(x-3)

to cancel out, leaving

\frac{x+3}{x+1}

. Multiplying this result by the second fraction,

\frac{x+1}{x}

, permits the cancellation of the common factor

(x+1)

, which simplifies the entire expression to the equivalent form

\frac{x+3}{x}

.

Adım Adım Çözüm

1

Factor the numerator and the denominator of the first rational expression:

\frac{x^2 - 9}{x^2 - 2x - 3}

.

The numerator factors as

(x - 3)(x + 3)

using the difference of squares identity. The denominator factors as

(x - 3)(x + 1)

by finding two numbers that multiply to

-3

and add to

-2

.

Factoring polynomials is necessary to identify and cancel common factors.

2

Simplify the first fraction by canceling the common factor

(x-3)

from both the numerator and denominator.

The first fraction simplifies to

\frac{x+3}{x+1}

.

Since

x > 3

, the term

x-3

is non-zero, making division by it valid.

3

Multiply the simplified first fraction by the second fraction:

\frac{x+3}{x+1} \cdot \frac{x+1}{x}

.

The common factor

(x+1)

in the numerator and denominator cancels out, resulting in

\frac{x+3}{x}

.

Canceling the common factor

(x+1)

yields the simplest equivalent expression.

Anahtar Kavram

Simplifying rational expressions by factoring polynomials and canceling common factors.

Soru 1487Soru →

For all $x > 3$ , the expression $\frac{x^3 - 2x^2 - 9x + 18}{x^2 - 5x + 6} - \frac{x^3 + 3x^2 - 4x - 12}{x^2 + 5x + 6}$ is equivalent to the constant $k$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of the constant

k

is 5.

Factoring the numerators by grouping yields

x^3 - 2x^2 - 9x + 18 = (x-3)(x+3)(x-2)

and

x^3 + 3x^2 - 4x - 12 = (x-2)(x+2)(x+3)

. Dividing out their respective denominators

(x-2)(x-3)

and

(x+2)(x+3)

leaves the simplified linear expressions

x+3

and

x-2

. Subtracting these yields

(x+3) - (x-2) = 5

.

Adım Adım Çözüm

1

Factor the numerator and denominator of the first rational expression and cancel common factors.

x + 3

To simplify the first fraction by reducing it to its lowest terms.

2

Factor the numerator and denominator of the second rational expression and cancel common factors.

x - 2

To simplify the second fraction by reducing it to its lowest terms.

3

Subtract the second simplified expression from the first simplified expression.

5

To find the constant value equivalent to the given difference of rational expressions, ensuring to distribute the negative sign to all parts of the subtracted binomial.

Anahtar Kavram

Simplifying rational expressions by factoring cubic polynomials by grouping and quadratic trinomials

Soru 1488Soru →

If the expression $\frac{6x^2 + 7x - 20}{2x - 3}$ is equivalent to $ax + b + \frac{c}{2x-3}$ for all $x \neq 1.5$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of

a + b + c

is 15.

By dividing the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

, we find that the quotient is

3x + 8

and the remainder is

4

. Thus, the expression can be rewritten as

3x + 8 + \frac{4}{2x-3}

. Comparing this to the given expression

ax + b + \frac{c}{2x-3}

, we obtain

a = 3

,

b = 8

, and

c = 4

. Their sum is

3 + 8 + 4 = 15

.

Adım Adım Çözüm

1

Set up the polynomial division of the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

.

Dividing

6x^2 + 7x - 20

by

2x - 3

.

To express the rational expression in terms of a polynomial quotient and a remainder.

2

Divide the first term of the numerator by the first term of the denominator to determine the first quotient term.

The first term is

3x

. Subtracting

3x(2x - 3)

from the numerator leaves

16x - 20

.

\frac{6x^2}{2x} = 3x

, and subtracting

6x^2 - 9x

from the polynomial leaves the next term to be divided.

3

Divide the leading term of the remaining expression by the leading term of the denominator to determine the constant term of the quotient.

The constant term is

8

. Subtracting

8(2x - 3)

from

16x - 20

leaves a remainder of

4

.

\frac{16x}{2x} = 8

, and subtracting

16x - 24

from

16x - 20

gives the final constant remainder.

4

Compare the quotient and remainder to the given form to identify

a

,

b

, and

c

.

a = 3

,

b = 8

, and

c = 4

.

The quotient is

3x + 8

and the remainder is

4

, which matches the form

ax + b + \frac{c}{2x-3}

.

5

Calculate the sum of

a

,

b

, and

c

.

3 + 8 + 4 = 15

.

The question asks for the value of

a + b + c

.

Anahtar Kavram

Polynomial division and rewrite of rational expressions

Soru 1489Soru →

In the field of paleoethnobotany, phytoliths—microscopic silica structures formed within plant tissues—are invaluable because they can survive for thousands of years after the surrounding organic material decays. By analyzing these durable microfossils, researchers can reconstruct ancient agricultural ______ identifying the exact plant species from phytolith shapes alone remains a significant challenge due to structural variations.

Which choice completes the text so that it conforms to the conventions of Standard English?

practices, however,

practices

practices; however,

practices, so

Cevabı ve açıklamayı göster

Cevap: practices; however,

Cevap

practices; however,

The sentence contains two independent clauses. The first independent clause is 'By analyzing these durable microfossils, researchers can reconstruct ancient agricultural practices,' and the second is 'identifying the exact plant species from phytolith shapes alone remains a significant challenge due to structural variations.' To connect two independent clauses, a semicolon must be used. Furthermore, because the second clause presents a contrast to the first, the conjunctive adverb 'however' is appropriate and must be followed by a comma.

Adım Adım Çözüm

1

Identify the clause boundaries in the sentence.

The sentence contains two independent clauses: 'By analyzing these durable microfossils, researchers can reconstruct ancient agricultural practices' and 'identifying the exact plant species from phytolith shapes alone remains a significant challenge due to structural variations.'

Determining where one independent clause ends and the next begins is necessary to select the correct punctuation and linking words.

2

Analyze the relationship between the two independent clauses.

The second clause presents a limitation or contrast to the first clause.

Understanding the logical connection helps choose the right transition word or conjunction.

3

Evaluate the choices to find the one that grammatically and logically connects the clauses.

The option with a semicolon followed by 'however' and a comma correctly joins the two independent clauses while expressing a contrast.

A semicolon is a standard way to link independent clauses, and 'however' is the correct conjunctive adverb for contrast.

Anahtar Kavram

Clause Boundaries and Linking

Soru 1490Soru →

A commercial cargo aircraft's total weight $W$ , in kilograms, is modeled as a linear function of the volume of fuel $F$ , in liters, in its fuel tanks. The equation modeling this relationship is:

W = 0.8F + 74,000

To comply with runway safety regulations at a destination airport, the aircraft's total weight must not exceed $85,000$ kilograms upon landing. The aircraft takes off with $18,000$ liters of fuel and consumes fuel at a constant rate of $1,700$ liters per hour of flight. What is the minimum number of hours the aircraft must fly before it can safely land?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The minimum number of hours the aircraft must fly is 2.5.

To satisfy runway safety regulations, the aircraft's weight must be at most

85,000

kg. According to the weight model

W = 0.8F + 74,000

, we set

0.8F + 74,000 \leq 85,000

, which simplifies to

0.8F \leq 11,000

. Dividing by the density coefficient

0.8

yields

F \leq 13,750

liters as the maximum amount of fuel the aircraft can contain at landing. Since the aircraft takes off with

18,000

liters of fuel and consumes

1,700

liters per hour, the fuel remaining after

t

hours is

18,000 - 1,700t

. To ensure the remaining fuel is less than or equal to

13,750

liters, we solve

18,000 - 1,700t \leq 13,750

, which simplifies to

4,250 \leq 1,700t

, or

t \geq 2.5

hours. Therefore, the minimum duration of the flight is 2.5 hours.

Adım Adım Çözüm

1

Formulate the weight limit inequality using the linear relationship.

0.8F + 74,000 \leq 85,000

The aircraft's weight

W

is given by

0.8F + 74,000

, where

0.8

represents the density of the fuel in kg/L and

74,000

represents the weight of the empty aircraft and its cargo. This total weight must not exceed

85,000

kg.

2

Calculate the maximum fuel capacity allowed at landing by solving the inequality.

F \leq 13,750

Subtracting

74,000

from both sides yields

0.8F \leq 11,000

. Dividing by

0.8

gives

F \leq 13,750

liters as the fuel ceiling for landing.

3

Use the fuel consumption rate to find the minimum flight time.

t \geq 2.5

With an initial fuel volume of

18,000

liters and a burn rate of

1,700

liters/hour, the fuel remaining after

t

hours is

18,000 - 1,700t

. Setting this expression to be at most

13,750

liters yields

18,000 - 1,700t \leq 13,750

. Subtracting

18,000

gives

-1,700t \leq -4,250

, and dividing by

-1,700

results in

t \geq 2.5

hours.

Anahtar Kavram

Interpreting linear coefficients in context and setting up linear inequalities with rates to model real-world constraints.

Tahmini Süre:2m 30s

Soru 1491Soru →

In the $xy$ -plane, the graphs of the linear function $3x + y = 6$ and the quadratic function $y = 3x^2 - 5x - 2$ intersect at two points. What is the sum of the $y$ -coordinates of these two intersection points?

10

\frac{2}{3}

14

2

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

The correct answer is 10. By rewriting the linear equation as

y = 6 - 3x

and substituting it into the quadratic equation, we get

6 - 3x = 3x^2 - 5x - 2

. Collecting all terms on one side yields the quadratic equation

3x^2 - 2x - 8 = 0

. Factoring this equation gives

(3x + 4)(x - 2) = 0

, which results in

x = 2

and

x = -\frac{4}{3}

. Substituting these

x

-values back into the linear equation gives the

y

-coordinates:

y = 6 - 3(2) = 0

and

y = 6 - 3(-\frac{4}{3}) = 10

. The sum of these

y

-coordinates is

0 + 10 = 10

.

Adım Adım Çözüm

1

Express

y

in terms of

x

using the linear equation.

y = 6 - 3x

Isolating

y

makes substitution into the quadratic equation straightforward.

2

Substitute the expression for

y

into the quadratic equation and set the equation to zero.

6 - 3x = 3x^2 - 5x - 2

, which simplifies to

3x^2 - 2x - 8 = 0

.

This forms a single quadratic equation in terms of

x

to find the

x

-coordinates of the intersection points.

3

Solve the quadratic equation by factoring.

(3x + 4)(x - 2) = 0

, yielding

x = 2

and

x = -\frac{4}{3}

.

Factoring determines the values of

x

at the points of intersection.

4

Substitute the

x

-values back into the linear equation

y = 6 - 3x

to find the corresponding

y

-coordinates.

For

x = 2

:

y = 6 - 3(2) = 0

. For

x = -\frac{4}{3}

:

y = 6 - 3(-\frac{4}{3}) = 10

.

This identifies the coordinates of the two intersection points as

(2, 0)

and

(-\frac{4}{3}, 10)

.

5

Calculate the sum of the

y

-coordinates.

0 + 10 = 10

The question asks for the sum of the

y

-coordinates of the two intersection points.

Anahtar Kavram

Solving a system of linear and quadratic equations using substitution.

Tahmini Süre:1m 30s

Soru 1492Soru →

A scientist studying a sample of a radioactive isotope determines that its mass decays exponentially. The mass of the isotope, in grams, is modeled by the function $M(t) = a \cdot b^t$ , where $t$ is the time, in days, since the study began, and $a$ and $b$ are constants. The table below shows the mass of the isotope for selected values of $t$ .

$t$ (days)	$M(t)$ (grams)
0	128
1	96
2	72

What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 0.75

Cevap

0.75

To find the constant

b

in the exponential decay function

M(t) = a \cdot b^t

, we can use the given table values. For

t = 0

,

M(0) = a \cdot b^0 = a = 128

. For

t = 1

,

M(1) = a \cdot b^1 = a \cdot b = 96

. Substituting

a = 128

into the second equation gives

128b = 96

. Dividing both sides by

128

yields

b = \frac{96}{128} = 0.75

(or

\frac{3}{4}

). We can verify this with

t = 2

:

M(2) = 128 \cdot (0.75)^2 = 128 \cdot 0.5625 = 72

, which matches the table.

Adım Adım Çözüm

1

Determine the initial value

a

using the data point for

t = 0

.

a = 128

Substituting

t = 0

into

M(t) = a \cdot b^t

gives

M(0) = a \cdot b^0 = a

. Since the table shows

M(0) = 128

,

a

must equal

128

.

2

Use the data point for

t = 1

to write an equation for

b

.

128b = 96

Substituting

t = 1

and

a = 128

into the function gives

M(1) = 128 \cdot b^1 = 128b

. The table shows

M(1) = 96

.

3

Solve the equation to find

b

.

b = 0.75

Dividing both sides of

128b = 96

by

128

yields

b = \frac{96}{128}

, which simplifies to

0.75

(or

\frac{3}{4}

).

Anahtar Kavram

Finding the decay factor (base) of an exponential function from a table of values.

Soru 1493Soru →

If $9^{2x + 1} = 27^{x - 3}$ , what is the value of $x$ ?

-5

-4

-11

10

Cevabı ve açıklamayı göster

Cevap: $-11$

Cevap

-11

To solve

9^{2x + 1} = 27^{x - 3}

, express 9 and 27 with a common base of 3:

(3^2)^{2x + 1} = (3^3)^{x - 3}

. Apply the power of a power rule to get

3^{2(2x + 1)} = 3^{3(x - 3)}

, which simplifies to

3^{4x + 2} = 3^{3x - 9}

. Equating the exponents gives

4x + 2 = 3x - 9

. Solving for

x

by subtracting

3x

and

2

from both sides yields

x = -11

. Therefore, the value of

x

is

-11

.

Adım Adım Çözüm

1

Rewrite both sides of the equation with a common base.

(3^2)^{2x+1} = (3^3)^{x-3}

Since 9 and 27 are both powers of 3 (

9 = 3^2

and

27 = 3^3

), expressing them with the same base allows the application of exponent rules.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

3^{2(2x+1)} = 3^{3(x-3)}

, which simplifies to

3^{4x+2} = 3^{3x-9}

Multiplying the inner exponent by the outer exponent simplifies the expression on both sides.

3

Set the exponents equal to each other.

4x + 2 = 3x - 9

Since the bases are equal and positive, the exponential expressions are equal if and only if their exponents are equal.

4

Solve the linear equation for

x

.

x = -11

Subtracting

3x

and

2

from both sides isolates the variable

x

.

Anahtar Kavram

Solving exponential equations by converting to a common base and equating exponents.

Soru 1494Soru →

A logistics company uses two types of boxes, small and large, to ship items. A shipment of $8$ small boxes and $5$ large boxes has a total weight of $180$ pounds. A second shipment of $6$ small boxes and $10$ large boxes has a total weight of $260$ pounds. What is the weight, in pounds, of one large box?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The weight of one large box is 20 pounds.

The correct weight of one large box is 20 pounds. By formulating the system of linear equations representing the total weight of each shipment (

8s + 5L = 180

and

6s + 10L = 260

), we can eliminate

L

by multiplying the first equation by 2, resulting in

16s + 10L = 360

. Subtracting the second equation from this gives

10s = 100

, which solves to

s = 10

. Substituting

s = 10

back into

8s + 5L = 180

yields

80 + 5L = 180

, which simplifies to

5L = 100

, meaning

L = 20

.

Adım Adım Çözüm

1

Set up a system of two linear equations based on the shipments.

Let

s

be the weight of a small box and

L

be the weight of a large box. The system of equations is:

\begin{aligned} 8s + 5L &= 180 \\ 6s + 10L &= 260 \end{aligned}

Translating the verbal statements into mathematical symbols represents the problem systemically.

2

Multiply the first equation by 2 to align the coefficients of

L

.

16s + 10L = 360

This sets up the variable

L

to have the same coefficient in both equations, allowing for elimination.

3

Subtract the second equation from the modified first equation to solve for

s

.

(16s + 10L) - (6s + 10L) = 360 - 260

10s = 100

s = 10

Subtracting eliminates the variable

L

, leaving a single-variable linear equation to solve.

4

Substitute

s = 10

back into the first equation to solve for

L

.

8(10) + 5L = 180

80 + 5L = 180

5L = 100

L = 20

Substituting the value of the solved variable determines the value of the remaining variable.

Anahtar Kavram

Solving systems of linear equations in two variables using elimination or substitution.

Soru 1495Soru →

While the class of marine organisms known as cephalopods possesses advanced problem-solving skills, ______ evolutionary history remains poorly understood due to a lack of fossilized remains.

Which choice completes the text so that it conforms to the conventions of Standard English?

their

its

it's

they're

Cevabı ve açıklamayı göster

Cevap: its

Cevap

its

The singular possessive pronoun 'its' correctly refers back to the singular collective noun antecedent 'class' and modifies 'evolutionary history'.

Adım Adım Çözüm

1

Identify the antecedent of the pronoun needed to modify 'evolutionary history'.

The antecedent is the singular noun phrase 'the class', not the plural noun 'cephalopods'.

The singular verb 'possesses' in the dependent clause confirms that 'the class' is the grammatical subject and antecedent.

2

Determine the correct grammatical number and case for the pronoun.

A singular possessive pronoun is required.

The pronoun must be singular to agree with 'class' and possessive to modify 'evolutionary history'.

3

Select the option that is singular, possessive, and spelled correctly without an apostrophe.

'its' is chosen over 'their', 'it's', and 'they're'.

'its' is the singular possessive pronoun, whereas 'it's' is a contraction of 'it is'.

Anahtar Kavram

Pronoun-Antecedent Agreement and Case

Soru 1496Soru →

In the $xy$ -plane, the graphs of the linear equation $y = 3x + k$ , where $k$ is a constant, and the quadratic equation $y = x^2 - 2x + 7$ intersect at two points. If the distance between the two points of intersection is $\sqrt{130}$ , what is the value of $k$ ?

1

-4

4

10

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct value of

k

is

4

. Equating the two equations yields the quadratic equation

x^2 - 5x + (7 - k) = 0

. Since the slope of the line is

3

, the difference in the y-coordinates of the intersection points is

3

times the difference in the x-coordinates. Applying the distance formula shows that the distance is equal to

\sqrt{10(x_2 - x_1)^2}

. Setting this equal to the given distance of

\sqrt{130}

yields

(x_2 - x_1)^2 = 13

. Substituting Vieta's relations (

x_1 + x_2 = 5

and

x_1 x_2 = 7 - k

) into the identity

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

results in

13 = 25 - 4(7 - k)

, which simplifies to

k = 4

.

Adım Adım Çözüm

1

Equate the linear and quadratic equations to set up the intersection condition.

x^2 - 2x + 7 = 3x + k \dots \implies x^2 - 5x + (7 - k) = 0

The x-coordinates of the intersection points are the roots of this quadratic equation.

2

Express the distance between the two points

(x_1, y_1)

and

(x_2, y_2)

in terms of the difference of their x-coordinates.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(x_2 - x_1)^2 + (3(x_2 - x_1))^2} = \sqrt{10(x_2 - x_1)^2}

Since both points lie on the line

y = 3x + k

, the difference in their y-coordinates is

y_2 - y_1 = 3(x_2 - x_1)

.

3

Use the given distance to find the value of

(x_2 - x_1)^2

.

10(x_2 - x_1)^2 = 130 \implies (x_2 - x_1)^2 = 13

The distance between the intersection points is given as

\sqrt{130}

.

4

Apply Vieta's formulas and the algebraic identity for the squared difference of the roots to solve for

k

.

x_1 + x_2 = 5

and

x_1 x_2 = 7 - k

. Substituting these into

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

gives

13 = 5^2 - 4(7 - k) \implies 13 = 25 - 28 + 4k \implies 4k = 16 \implies k = 4

.

Vieta's formulas relate the coefficients of the quadratic equation to the sum and product of its roots.

Anahtar Kavram

Solving systems of linear and quadratic equations and finding constants based on geometric constraints.

Soru 1497Soru →

If $x^2 - 9x + 18 = 0$ and $x > 4$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is 6.

Factoring the equation

x^2 - 9x + 18 = 0

gives

(x - 3)(x - 6) = 0

, which yields the solutions

x = 3

and

x = 6

. Applying the constraint

x > 4

, the only valid solution is 6.

Adım Adım Çözüm

1

Factor the quadratic equation

x^2 - 9x + 18 = 0

.

(x - 3)(x - 6) = 0

Finding two numbers that multiply to 18 and add to -9 allows us to write the quadratic expression in its factored form.

2

Solve for the roots of the equation.

x = 3

or

x = 6

By the zero product property, setting each factor to zero yields the possible solutions for the equation.

3

Apply the given constraint

x > 4

.

x = 6

Since the question specifies that

x

must be greater than 4, the root

x = 3

is discarded, leaving

x = 6

as the only valid solution.

Anahtar Kavram

Solving quadratic equations by factoring and applying constraints.

Soru 1498Soru →

The quadratic function $f$ is defined by $f(x) = (x - 5)(x - 11)$ . In the $xy$ -plane, the graph of $y = f(x)$ is a parabola with vertex $(h, k)$ . What is the value of $h$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of

h

is

8

.

The given quadratic function is in factored form,

f(x) = (x - 5)(x - 11)

, which shows that the

x

-intercepts of the graph are at

x = 5

and

x = 11

. Because a parabola is symmetric, the

x

-coordinate of its vertex,

h

, is located exactly halfway between the two

x

-intercepts. Calculating the average of the

x

-intercepts gives

h = \frac{5 + 11}{2} = 8

.

Adım Adım Çözüm

1

Find the

x

-intercepts of the parabola.

The

x

-intercepts are

x = 5

and

x = 11

.

The

x

-intercepts of a quadratic function in the form

f(x) = (x - r_1)(x - r_2)

occur at

x = r_1

and

x = r_2

because setting

f(x) = 0

yields these roots.

2

Calculate the

x

-coordinate of the vertex,

h

.

h = 8

Since a parabola is symmetric, its vertex lies on the axis of symmetry, which is the vertical line halfway between the

x

-intercepts. The average of

5

and

11

is

\frac{5 + 11}{2} = 8

.

Anahtar Kavram

Finding the vertex of a parabola from its factored form

Soru 1499Soru →

At a local coffee shop, the remaining weight of coffee beans $C$ , in pounds, after preparing $n$ cups of espresso is modeled by the equation $C = 80 - 0.04n$ . What is the best interpretation of the number $0.04$ in this context?

The weight of coffee beans, in pounds, used to prepare each cup of espresso

The initial weight of the coffee beans, in pounds, before any cups of espresso are prepared

The number of cups of espresso that can be prepared per pound of coffee beans

The weight of coffee beans, in pounds, remaining in the shop after preparing

80

cups of espresso

Cevabı ve açıklamayı göster

Cevap: The weight of coffee beans, in pounds, used to prepare each cup of espresso

Cevap

The weight of coffee beans, in pounds, used to prepare each cup of espresso

The correct answer is the option stating that

0.04

is the weight of coffee beans, in pounds, used to prepare each cup of espresso. In the linear relationship

C = 80 - 0.04n

, the slope is

-0.04

, which represents the rate of change of the remaining weight of coffee beans with respect to the number of cups of espresso prepared. A rate of

-0.04

pounds per cup means that for every cup prepared, the remaining coffee beans decrease by

0.04

pounds, indicating that

0.04

pounds of beans are consumed per cup.

Adım Adım Çözüm

1

Analyze the structure of the linear equation

C = 80 - 0.04n

.

The equation is in the slope-intercept form

y = mx + b

, where the dependent variable is

C

(remaining coffee beans in pounds), the independent variable is

n

(number of cups of espresso), the constant (y-intercept) is

80

, and the slope (coefficient of

n

) is

-0.04

.

Identifying the components of the linear equation helps in assigning their real-world contextual meanings.

2

Interpret the meaning of the slope in the context of the variables.

The slope of

-0.04

represents the change in the remaining coffee beans (

C

) for every

1

-unit increase in the number of cups of espresso (

n

). This means the remaining weight decreases by

0.04

pounds per cup.

The slope of a linear model shows the constant rate of change of the dependent variable per unit of the independent variable.

3

Relate the rate of decrease to the options.

A decrease of

0.04

pounds of remaining beans per cup of espresso means that

0.04

pounds of coffee beans are used to prepare each cup of espresso.

Translating the decrease rate into consumption rate matches the physical scenario described.

Anahtar Kavram

Interpreting Linear Relationships in Context

Tahmini Süre:45s

Soru 1500Soru →

A wildlife biologist models the population of a certain bird species in a nature reserve using the function $P(t) = 120(1.05)^t$ , where $P(t)$ represents the number of birds $t$ years after the study began. What does the number $120$ represent in this model?

The initial number of birds in the reserve when the study began

The number of birds added to the reserve each year

The percent by which the bird population increases each year

The number of years it takes for the bird population to double

Cevabı ve açıklamayı göster

Cevap: The initial number of birds in the reserve when the study began

Cevap

The initial number of birds in the reserve when the study began

In the exponential function model

P(t) = a \cdot b^t

, the coefficient

a

is the value of the function at

t = 0

. Evaluating

P(0)

yields

120(1.05)^0 = 120

, which corresponds to the initial number of birds in the reserve when the study began.

Adım Adım Çözüm

1

Identify the standard form of the exponential growth function

P(t) = a \cdot b^t

, where

a

represents the initial value and

b

represents the growth factor.

This helps map the given constants to their mathematical meanings.

2

Substitute

t = 0

to find the starting population

P(0) = 120(1.05)^0 = 120 \cdot 1 = 120

.

Finding the value at

t = 0

defines the initial state of the model.

3

Interpret the initial value in the context of the problem

The number

120

represents the number of birds present at the start of the study.

This links the mathematical result to the real-world scenario.

Anahtar Kavram

Interpreting components of an exponential function in context

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