Tüm alıştırma soruları

Soru 181Soru →

A circle and a line are graphed in the $xy$ -plane. The circle is defined by the equation $x^2 + y^2 = 25$ , and the line is defined by the equation $y = 3$ . If the line intersects the circle at the point $(x, 3)$ , where $x > 0$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is 4.

Substituting

y = 3

into the circle equation

x^2 + y^2 = 25

yields

x^2 + 3^2 = 25

. Simplifying this gives

x^2 + 9 = 25

, which simplifies to

x^2 = 16

. Taking the square root of both sides gives

x = 4

or

x = -4

. Since it is given that

x > 0

, the value of

x

must be

4

.

Adım Adım Çözüm

1

Substitute the value of

y = 3

into the circle's equation.

x^2 + 3^2 = 25

Since the line is

y = 3

, any point of intersection must satisfy this y-coordinate. Substituting it into the circle's equation allows us to solve for the x-coordinate.

2

Simplify the equation and isolate

x^2

.

x^2 = 16

Squaring

3

gives

9

, and subtracting

9

from both sides of the equation

x^2 + 9 = 25

isolates

x^2

.

3

Solve for

x

and apply the constraint

x > 0

.

x = 4

Taking the square root of both sides of

x^2 = 16

gives

x = 4

or

x = -4

. Since the problem states

x > 0

, the only valid solution is

4

.

Anahtar Kavram

Solving systems of nonlinear equations using substitution

Soru 182Soru →

An architect is tracking the height of a new skyscraper under construction. The total height $H$ , in feet, of the building $w$ weeks after construction of the main frames began is modeled by the equation $H = 120 + 15w$ . According to the model, by how many feet does the height of the building increase each week?

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Cevap: 15

Cevap

The height of the building increases by 15 feet each week.

In the linear model

H = 120 + 15w

, the variable

H

represents the total height of the building in feet and the variable

w

represents the number of weeks since construction began. The coefficient of the variable

w

, which is

15

, is the slope of the equation and represents the constant rate of change of the height per week. Therefore, the height of the building increases by

15

feet each week.

Adım Adım Çözüm

1

Analyze the linear model equation.

The equation

H = 120 + 15w

represents the total height

H

as a function of the number of weeks

w

, where

15

is the coefficient of the variable

w

.

To determine what each part of the linear equation represents in the given context.

2

Interpret the coefficient of the independent variable

w

.

The coefficient of

w

is the slope of the linear equation, which is

15

.

The slope represents the constant rate of change, which is the weekly increase in height.

Anahtar Kavram

Interpreting the slope of a linear equation in context

Soru 183Soru →

If the expression $\frac{4x^2 + kx - 7}{2x - 1}$ is equivalent to $2x + 5 - \frac{2}{2x - 1}$ for all $x \neq \frac{1}{2}$ , where $k$ is a constant, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The correct answer is 8. Combining the expression

2x + 5 - \frac{2}{2x - 1}

into a single fraction requires finding a common denominator of

2x - 1

. Multiplying the linear term

2x + 5

by

\frac{2x - 1}{2x - 1}

and subtracting 2 yields

\frac{(2x + 5)(2x - 1) - 2}{2x - 1}

. Expanding and simplifying the numerator gives

4x^2 + 8x - 7

. Equating this to the numerator of the original expression,

4x^2 + kx - 7

, shows that the coefficient of the linear term,

k

, must be equal to 8.

Adım Adım Çözüm

1

Multiply the linear expression by the denominator to prepare for combining the terms.

(2x + 5)(2x - 1) = 4x^2 + 8x - 5

To combine all terms under a single common denominator of

2x - 1

, the non-fractional terms must be multiplied by the denominator.

2

Subtract the numerator of the fractional term from the expanded product.

(4x^2 + 8x - 5) - 2 = 4x^2 + 8x - 7

This completes the subtraction of the fraction over the common denominator, resulting in a single rational expression.

3

Compare the resulting numerator to the numerator of the original expression to find the value of the constant.

k = 8

For the two rational expressions to be equivalent, their numerators must be equal for all values of

x

. Thus, the coefficient of

x

in both expressions must match.

Anahtar Kavram

Equivalence of rational expressions through finding a common denominator

Soru 184Soru →

A subscription-based meal kit service charges a monthly membership fee of $\$35$ plus $\$8.50$ per meal. A non-member can purchase the same meals for $\$12.00$ each, but must pay a flat monthly delivery fee of $\$7$ . How many meals must be purchased in a month for the total monthly cost for a member to be equal to the total monthly cost for a non-member?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The total monthly cost is equal for a member and a non-member when 8 meals are purchased in a month.

The correct answer of 8 is found by setting the member cost expression,

35 + 8.5m

, equal to the non-member cost expression,

12m + 7

, and solving for the number of meals,

m

.

Adım Adım Çözüm

1

Represent the total monthly cost for a member and a non-member using equations where

m

is the number of meals purchased.

Member cost is represented by

35 + 8.50m

, and non-member cost is represented by

12.00m + 7

.

To set up an algebraic representation of the cost structures.

2

Equate the two cost expressions to find the number of meals where the costs are equal.

35 + 8.50m = 12.00m + 7

To find the value of

m

that makes both costs equivalent.

3

Isolate the variable

m

on one side of the equation.

3.50m = 28

Subtracting

8.50m

and

7

from both sides simplifies the equation by keeping variable terms on one side and constants on the other.

4

Divide both sides of the equation by

3.50

to solve for

m

.

m = 8

Dividing

28

by

3.50

yields the final number of meals.

Anahtar Kavram

Solving linear equations in one variable with variable terms on both sides of the equation

Tahmini Süre:1m 30s

Soru 185Soru →

A bounded region in the $xy$ -plane is defined by the following system of inequalities:

\begin{aligned} y &\ge \frac{1}{2}x + 2 \\ y &\le 2x - 1 \\ y &\le -x + 11 \end{aligned}

What is the maximum $y$ -coordinate of any point within this region?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The maximum y-coordinate of any point in the bounded region is 7.

The solution region is a triangle with vertices at

(2, 3)

,

(4, 7)

, and

(6, 5)

. Since all inequalities are less-than-or-equal-to or greater-than-or-equal-to, the boundary points are included in the solution set. The maximum y-value occurs at the vertex

(4, 7)

, which gives a maximum y-coordinate of 7.

Adım Adım Çözüm

1

Determine the equations of the boundary lines.

The boundary equations are

y = \frac{1}{2}x + 2

,

y = 2x - 1

, and

y = -x + 11

.

These equations represent the boundaries of the system of inequalities.

2

Find the intersection point of

y = 2x - 1

and

y = \frac{1}{2}x + 2

.

Solving

2x - 1 = \frac{1}{2}x + 2

yields

x = 2

, which gives

y = 3

. The intersection is

(2, 3)

.

This is one vertex of the bounded region.

3

Find the intersection point of

y = 2x - 1

and

y = -x + 11

.

Solving

2x - 1 = -x + 11

yields

x = 4

, which gives

y = 7

. The intersection is

(4, 7)

.

This is the second vertex of the bounded region.

4

Find the intersection point of

y = -x + 11

and

y = \frac{1}{2}x + 2

.

Solving

-x + 11 = \frac{1}{2}x + 2

yields

x = 6

, which gives

y = 5

. The intersection is

(6, 5)

.

This is the third vertex of the bounded region.

5

Compare the y-coordinates of the three vertices of the bounded region.

The y-coordinates of the vertices

(2, 3)

,

(4, 7)

, and

(6, 5)

are 3, 7, and 5, respectively. The maximum value is 7.

The maximum value of a linear coordinate over a bounded convex region occurs at one of its vertices.

Anahtar Kavram

Finding the vertices of a bounded region defined by a system of linear inequalities and optimizing a coordinate value.

Soru 186Soru →

A system of equations consists of the equations $y = -x^2 + 2x + 7$ and $y = 6x + k$ , where $k$ is a constant. If the system has two distinct real solutions, what is the greatest integer value of $k$ ?

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Cevap: 10

Cevap

The correct answer is 10. The greatest integer value of the constant that allows the system to have two distinct real solutions is 10.

To find the number of solutions to the system, equate the two equations:

-x^2 + 2x + 7 = 6x + k

. Rearranging this equation into standard quadratic form gives

x^2 + 4x + (k - 7) = 0

. For the system to have two distinct real solutions, the discriminant of this quadratic equation must be strictly greater than zero. The discriminant is calculated as

b^2 - 4ac = 4^2 - 4(1)(k - 7) = 16 - 4k + 28 = 44 - 4k

. Setting this greater than zero yields

44 - 4k > 0

, which simplifies to

k < 11

. The greatest integer value of

k

that is strictly less than 11 is 10.

Adım Adım Çözüm

1

Equate the expressions for

y

from both equations.

-x^2 + 2x + 7 = 6x + k

At the points of intersection, the

y

-values of both equations must be equal.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 + 4x + (k - 7) = 0

Standard form is required to calculate the discriminant of the quadratic equation.

3

Write the expression for the discriminant

\Delta = b^2 - 4ac

using the coefficients from the quadratic equation.

\Delta = 4^2 - 4(1)(k - 7) = 44 - 4k

The discriminant determines the number of real solutions to the quadratic equation.

4

Set the discriminant to be strictly greater than 0 and solve the inequality for

k

.

44 - 4k > 0 \implies k < 11

For the system to have two distinct real solutions, the discriminant must be positive.

5

Determine the greatest integer value of

k

that satisfies the inequality

k < 11

.

10

The largest integer strictly less than 11 is 10.

Anahtar Kavram

Using the discriminant of a quadratic equation derived from a nonlinear system to determine the number of real solutions.

Soru 187Soru →

For all $x > 0$ , the expression $\left( \frac{x^{4/3} + 4x^{2/3} + 16}{x^{2/3} + 2x^{1/3} + 4} + 2x^{1/3} \right)^3 - x^2 - 12x^{4/3} - 48x^{2/3}$ is equivalent to a constant $C$ . What is the value of $C$ ?

Cevabı ve açıklamayı göster

Cevap: 64

Cevap

The constant value is 64.

The expression inside the parentheses simplifies to

x^{2/3} + 4

after factoring the numerator as

(x^{2/3} + 2x^{1/3} + 4)(x^{2/3} - 2x^{1/3} + 4)

and canceling the common factor in the denominator. Cubing

x^{2/3} + 4

yields

x^2 + 12x^{4/3} + 48x^{2/3} + 64

. Subtracting the remaining terms

x^2 + 12x^{4/3} + 48x^{2/3}

from this expansion results in the constant value 64.

Adım Adım Çözüm

1

Substitute

u = x^{1/3}

into the expression to simplify the fractional exponents.

The terms become

x^{1/3} = u

,

x^{2/3} = u^2

,

x^{4/3} = u^4

, and

x^2 = u^6

. The expression inside the parentheses is rewritten as

\frac{u^4 + 4u^2 + 16}{u^2 + 2u + 4} + 2u

.

Using a temporary variable

u

simplifies the algebraic factoring and manipulation of terms with fractional exponents.

2

Factor the numerator

u^4 + 4u^2 + 16

by completing the square.

u^4 + 4u^2 + 16 = (u^2 + 4)^2 - 4u^2 = (u^2 + 2u + 4)(u^2 - 2u + 4)

.

Expressing the quartic polynomial as a difference of squares allows it to be factored into two quadratic polynomials.

3

Simplify the rational expression and add

2u

.

\frac{(u^2 + 2u + 4)(u^2 - 2u + 4)}{u^2 + 2u + 4} + 2u = (u^2 - 2u + 4) + 2u = u^2 + 4

.

Canceling the common factor

u^2 + 2u + 4

in the numerator and denominator simplifies the expression inside the parentheses to

u^2 + 4

.

4

Substitute

u = x^{1/3}

back into

u^2 + 4

and cube the expression.

(x^{2/3} + 4)^3 = (x^{2/3})^3 + 3(x^{2/3})^2(4) + 3(x^{2/3})(16) + 64 = x^2 + 12x^{4/3} + 48x^{2/3} + 64

.

Applying the binomial expansion formula

(A + B)^3 = A^3 + 3A^2B + 3AB^2 + B^3

expands the cubed expression.

5

Subtract the remaining terms from the expanded expression.

(x^2 + 12x^{4/3} + 48x^{2/3} + 64) - x^2 - 12x^{4/3} - 48x^{2/3} = 64

.

Subtracting the variable terms cancels them out entirely, leaving the constant value 64.

Anahtar Kavram

Factoring quartic polynomials using the difference of squares and simplifying rational expressions with fractional exponents.

Soru 188Soru →

A landscaping company is planting $x$ maple trees and $y$ pine trees in a park. The number of trees of each type must satisfy the system of inequalities below:

\begin{aligned} 30x + 40y &\le 360 \\ x + y &\ge 10 \\ x &\ge 4 \end{aligned}

What is the maximum number of pine trees the company can plant?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The maximum number of pine trees the company can plant is 6.

To find the maximum number of pine trees,

y

, we look at the boundary constraints. The constraint

x \ge 4

states that at least

4

maple trees must be planted. Since planting fewer maple trees leaves more of the budget for pine trees, we minimize

x

by setting

x = 4

. Substituting this value into the budget inequality

30x + 40y \le 360

gives

120 + 40y \le 360

. Solving for

y

yields

40y \le 240

, which simplifies to

y \le 6

. We then verify that the solution

(4, 6)

satisfies the total tree constraint

x + y \ge 10

, which it does since

4 + 6 = 10

. Thus, the maximum number of pine trees is 6.

Adım Adım Çözüm

1

Substitute the minimum possible value of

x

into the first inequality.

Since

x \ge 4

, the smallest possible value for

x

is

4

. Substituting

x = 4

into

30x + 40y \le 360

yields:

30(4) + 40y \le 360

120 + 40y \le 360

To maximize the value of

y

under the resource constraint, we must minimize the value of

x

.

2

Solve the inequality for

y

.

40y \le 240

y \le 6

This establishes that the maximum possible value for

y

based on the budget constraint is

6

.

3

Verify that

(4, 6)

satisfies all inequalities in the system.

Checking the second inequality:

x + y \ge 10 \implies 4 + 6 = 10 \ge 10

, which is true. The third inequality

x \ge 4 \implies 4 \ge 4

is also true.

A coordinate pair must satisfy all inequalities in the system to be a valid solution.

Anahtar Kavram

To find the maximum value of a variable in a system of inequalities with constraints, analyze the boundary lines and the intersection points of the feasible region.

Soru 189Soru →

In the $xy$ -plane, the graph of a line $l$ passes through the points $(0, 1)$ and $(3, 5)$ . If another point on line $l$ has coordinates $(t, 9)$ , what is the value of $t$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

t

is

6

.

The slope of line

l

is

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

. Using the y-intercept

(0, 1)

, the equation of the line is

y = \frac{4}{3}x + 1

. Setting

y = 9

gives

9 = \frac{4}{3}t + 1

. Subtracting 1 from both sides gives

8 = \frac{4}{3}t

. Multiplying both sides by

\frac{3}{4}

yields

t = 6

.

Adım Adım Çözüm

1

Calculate the slope of line

l

using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the points

(0, 1)

and

(3, 5)

.

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

The slope of a line represents its constant rate of change and is needed to determine the line's equation.

2

Write the equation of the line in slope-intercept form,

y = mx + b

, using the slope

m = \frac{4}{3}

and the y-intercept

b = 1

(from the point

(0, 1)

).

y = \frac{4}{3}x + 1

The slope-intercept equation defines the relationship between the

x

- and

y

-coordinates of any point on the line.

3

Substitute the point

(t, 9)

into the line's equation and solve for

t

.

9 = \frac{4}{3}t + 1 \implies 8 = \frac{4}{3}t \implies t = 6

Since the point lies on the line, its coordinates must satisfy the line's equation.

Anahtar Kavram

Determining the equation of a linear function from a graph or points and evaluating it for a given value.

Soru 190Soru →

For the system of equations shown below,

(x, y)

is the unique solution:

\begin{aligned} \frac{3}{4}(2x - y) &= \frac{2}{3}(x + 2y) \\ \frac{1}{2}(x - y) + \frac{5}{6}(2x + y) &= 11.5 \end{aligned}

What is the value of

x + y

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x + y

is 7.

To find the value of

x + y

, we first simplify the first equation by multiplying both sides by the least common multiple of

3

and

4

, which is

12

. This yields

9(2x - y) = 8(x + 2y)

, which simplifies to

18x - 9y = 8x + 16y

, and further simplifies to

10x = 25y

, or

x = 2.5y

. Next, we substitute

x = 2.5y

into the second equation:

\frac{1}{2}(2.5y - y) + \frac{5}{6}(2(2.5y) + y) = 11.5

. Simplifying the terms gives

0.75y + 5y = 11.5

, or

5.75y = 11.5

, which yields

y = 2

. Substituting

y = 2

back into

x = 2.5y

gives

x = 5

. Thus, the value of

x + y

is

5 + 2 = 7

.

Adım Adım Çözüm

1

Clear the fractions in the first equation by multiplying by the least common multiple of the denominators.

10x = 25y

, which simplifies to

x = 2.5y

.

To express one variable in terms of the other for substitution.

2

Substitute the expression for

x

into the second equation and solve for

y

.

y = 2

To find the numerical value of one of the variables.

3

Substitute the value of

y

back into the simplified first equation to find

x

.

x = 5

To find the numerical value of the remaining variable.

4

Add the values of

x

and

y

to find

x + y

.

7

To answer the specific question asked.

Anahtar Kavram

Solving systems of linear equations using algebraic simplification and substitution

Soru 191Soru →

A landscaping service uses a water tank to irrigate lawns. The volume of water, $V$ , in gallons, remaining in the tank after $n$ lawns have been irrigated is modeled by the equation $V = 850 - 25n$ . According to the model, by how many gallons does the volume of water in the tank decrease for each lawn that is irrigated?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The correct answer is 25, which represents the decrease in the volume of water in the tank, in gallons, for each lawn irrigated.

In the linear equation

V = 850 - 25n

, the coefficient of

n

is

-25

. This coefficient represents the rate of change of the volume of water with respect to the number of lawns irrigated. The negative sign shows that the volume is decreasing, and the magnitude, 25, indicates that the volume decreases by 25 gallons for each lawn irrigated.

Adım Adım Çözüm

1

Identify the coefficient of the variable

n

in the equation

V = 850 - 25n

.

The coefficient of

n

is

-25

.

The coefficient of the independent variable in a linear equation represents the rate of change of the dependent variable.

2

Interpret the coefficient in terms of the real-world context.

The coefficient

-25

means the volume of water decreases by 25 gallons for each lawn irrigated.

The negative sign indicates a decrease, and the magnitude represents the amount of change per unit.

Anahtar Kavram

Interpreting the slope of a linear relationship in context.

Soru 192Soru →

An environmental cleanup crew is removing a contaminant from a soil site. The remaining mass of the contaminant, $C$ , in kilograms, after $d$ days of treatment is modeled by the equation $C = 400 - p d$ , where $p$ is the daily removal rate, in kilograms per day, under the original protocol. Under a new treatment protocol, the daily removal rate is increased by $25\%$ , and the treatment time required to completely remove the contaminant is reduced by $8$ days. What was the daily removal rate, in kilograms per day, under the original protocol?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The daily removal rate under the original protocol was 10 kilograms per day.

Under the original protocol, complete removal of the

400

kg contaminant occurs when

C = 0

, giving a duration of

d = \frac{400}{p}

days. Under the new protocol, the removal rate increases by

25\%

to

1.25p

, and the duration is reduced by

8

days to

\frac{400}{p} - 8

. Because the total mass removed must still equal

400

kg, we write the equation

(1.25p)\left(\frac{400}{p} - 8\right) = 400

. Distributing

1.25p

yields

500 - 10p = 400

. Solving for

p

gives

10p = 100

, which simplifies to

p = 10

kilograms per day.

Adım Adım Çözüm

1

Set

C = 0

in the original equation to represent complete removal.

0 = 400 - p d \implies d = \frac{400}{p}

Complete removal of the contaminant means that the remaining mass

C

is

0

kilograms.

2

Express the new daily removal rate and the new treatment duration using the given percentage increase and day reduction.

p_{\text{new}} = 1.25p

and

d_{\text{new}} = d - 8 = \frac{400}{p} - 8

The new protocol increases the daily removal rate by

25\%

and reduces the total treatment time by

8

days.

3

Set up the equation for complete removal under the new protocol using the new rate and duration.

400 - (1.25p) \left(\frac{400}{p} - 8\right) = 0

The total initial contaminant mass of

400

kilograms must be completely removed by the new daily rate over the new duration.

4

Solve the equation for

p

.

1.25p \left(\frac{400}{p} - 8\right) = 400 \implies 500 - 10p = 400 \implies 10p = 100 \implies p = 10

Distribute

1.25p

into the parentheses to eliminate the fraction, then isolate the variable

p

.

Anahtar Kavram

Interpreting the rate (slope) and intercepts of a linear relationship in context, and modeling variations of those parameters.

Soru 193Soru →

In a certain video game, players earn points for completing quests and defeating bosses. Completing a quest earns $q$ points, and defeating a boss earns $b$ points. Leo completed $5$ quests and defeated $3$ bosses, earning a total of $250$ points. Maya completed $7$ quests and defeated $2$ bosses, earning a total of $240$ points. What is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of

q

is

20

.

To find the value of

q

, we translate the given scenarios into a system of two linear equations:

5q + 3b = 250

and

7q + 2b = 240

. Multiplying the first equation by

2

gives

10q + 6b = 500

. Multiplying the second equation by

3

gives

21q + 6b = 720

. Subtracting the first equation from the second yields

(21q - 10q) + (6b - 6b) = 720 - 500

, which simplifies to

11q = 220

. Dividing both sides by

11

results in

q = 20

.

Adım Adım Çözüm

1

Set up the system of linear equations from the given information.

5q + 3b = 250

and

7q + 2b = 240

To represent the points earned by Leo and Maya mathematically.

2

Multiply the first equation by

2

and the second equation by

3

.

10q + 6b = 500

and

21q + 6b = 720

To make the coefficients of

b

equal so they can be eliminated.

3

Subtract the first modified equation from the second modified equation.

11q = 220

To eliminate

b

and solve for

q

directly.

4

Divide both sides of the equation by

11

.

q = 20

To find the number of points earned per completed quest.

Anahtar Kavram

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

Soru 194Soru →

In the quadratic equation $x^2 - kx + 9 = 0$ , $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

For a quadratic equation

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant must equal zero (

b^2 - 4ac = 0

). In the equation

x^2 - kx + 9 = 0

, the coefficients are

a = 1

,

b = -k

, and

c = 9

. Setting the discriminant to zero gives

(-k)^2 - 4(1)(9) = 0

, which simplifies to

k^2 - 36 = 0

. Solving this equation yields

k^2 = 36

, so

k = 6

or

k = -6

. Since

k

is a positive constant, we reject the negative root, leaving

k = 6

.

Adım Adım Çözüm

1

Identify the condition for a quadratic equation to have exactly one real solution.

The discriminant of the quadratic equation must be equal to 0, which is represented by the formula

b^2 - 4ac = 0

.

The discriminant determines the number of real solutions of a quadratic equation. If the discriminant is 0, there is exactly one real solution.

2

Identify the coefficients

a

,

b

, and

c

of the given equation

x^2 - kx + 9 = 0

.

a = 1

,

b = -k

, and

c = 9

.

These coefficients are required to compute the value of the discriminant.

3

Set the discriminant equal to 0 and simplify the equation.

(-k)^2 - 4(1)(9) = 0 \Rightarrow k^2 - 36 = 0

.

Substituting the coefficients into the discriminant formula sets up the algebraic relationship to find

k

.

4

Solve for the positive constant

k

.

k^2 = 36 \Rightarrow k = 6

or

k = -6

. Since

k

must be positive,

k = 6

.

Solving the equation gives two possible values, but the negative solution is discarded because the problem specifies

k

is a positive constant.

Anahtar Kavram

Discriminant of a quadratic equation

Alternatif Yöntem

Alternatively, a quadratic equation has exactly one real solution if it can be written as a perfect square trinomial in the form

(x - d)^2 = 0

, which expands to

x^2 - 2dx + d^2 = 0

. Comparing this with

x^2 - kx + 9 = 0

, we get

d^2 = 9

and

2d = k

. Since

d^2 = 9

,

d

can be

3

or

-3

. Given that

k

is positive and

k = 2d

,

d

must also be positive, meaning

d = 3

. Substituting this back gives

k = 2(3) = 6

.

Tahmini Süre:45s

Soru 195Soru →

A population of bacteria doubles every 3 hours. If the initial population of the bacteria is 500, what is the population of the bacteria after 9 hours?

Cevabı ve açıklamayı göster

Cevap: 4000

Cevap

The population of the bacteria after 9 hours is 4,000.

The final population is calculated using the formula

P(t) = P_0 \times 2^{t/d}

, where

P_0

is the initial population of 500,

d

is the doubling period of 3 hours, and

t

is the total time of 9 hours. Evaluating this gives

P(9) = 500 \times 2^{9/3} = 500 \times 2^3 = 500 \times 8 = 4000

.

Adım Adım Çözüm

1

Identify the initial population (

P_0

), doubling time (

d

), and total time (

t

).

P_0 = 500

,

d = 3

, and

t = 9

.

These parameters are required to set up the exponential growth model.

2

Calculate the number of doubling periods.

The number of doubling periods is

\frac{9}{3} = 3

.

The population doubles once for every 3-hour interval.

3

Calculate the final population using the exponential growth formula.

500 \times 2^3 = 500 \times 8 = 4000

.

Applying the 3 doubling cycles to the initial population of 500 yields the final population.

Anahtar Kavram

Exponential Growth Model

Tahmini Süre:1m 0s

Soru 196Soru →

In the system of equations below, $k$ is a positive constant.

x - y = k

x^2 - 3xy + y^2 = 5

If the system has exactly one real solution $(x, y)$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

is 2.

Substituting

y = x - k

into the second equation yields

-x^2 + kx + k^2 - 5 = 0

, which can be rewritten in standard form as

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

. For this quadratic equation to have exactly one real solution, its discriminant must equal 0:

b^2 - 4ac = (-k)^2 - 4(1)(5 - k^2) = 5k^2 - 20 = 0

. Solving

5k^2 - 20 = 0

gives

k^2 = 4

, and since

k

must be positive,

k = 2

.

Adım Adım Çözüm

1

Rearrange the first equation to express

y

in terms of

x

.

y = x - k

This allows for substitution into the second equation to eliminate

y

.

2

Substitute

y = x - k

into the second equation and expand.

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

To create a single quadratic equation in terms of

x

.

3

Multiply by

-1

to write the quadratic equation in standard form

ax^2 + bx + c = 0

.

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

Standard form makes it easier to identify the coefficients

a = 1

,

b = -k

, and

c = 5 - k^2

.

4

Set the discriminant

b^2 - 4ac

equal to 0.

(-k)^2 - 4(1)(5 - k^2) = 0

A quadratic equation has exactly one real solution if and only if its discriminant is zero.

5

Simplify the discriminant equation and solve for

k

.

5k^2 - 20 = 0 \implies k^2 = 4 \implies k = 2

(since

k

must be positive)

To find the positive constant

k

that satisfies the condition.

Anahtar Kavram

Determining the number of solutions to a nonlinear system by substituting and setting the discriminant of the resulting quadratic equation to zero.

Alternatif Yöntem

Alternatively, one can rewrite the second equation by grouping:

x^2 - 3xy + y^2 = (x - y)^2 - xy = 5

. Since

x - y = k

, we have

k^2 - xy = 5

, so

xy = k^2 - 5

. We now have a system of

x - y = k

and

xy = k^2 - 5

. Substituting

y = x - k

gives

x(x - k) = k^2 - 5

, leading to

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

, which can be solved using the discriminant as shown in the primary method.

Tahmini Süre:2m 30s

Soru 197Soru →

For all $x > 0$ and $y > 0$ , the expression $\frac{(x^2y^3)^a}{(xy^2)^2}$ is equivalent to $x^6y^8$ , where $a$ is a constant. What is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Applying the rules of exponents, the expression

\frac{(x^2y^3)^a}{(xy^2)^2}

simplifies to

\frac{x^{2a}y^{3a}}{x^2y^4} = x^{2a-2}y^{3a-4}

. Setting this equal to the equivalent expression

x^6y^8

gives the system of equations

2a - 2 = 6

and

3a - 4 = 8

. Solving either equation yields

a = 4

.

Adım Adım Çözüm

1

Apply the power rule of exponents to the numerator and denominator.

Numerator:

x^{2a}y^{3a}

, Denominator:

x^2y^4

To expand the parentheses by multiplying the outer exponent with the inner exponents.

2

Apply the quotient rule of exponents to divide the numerator by the denominator.

x^{2a-2}y^{3a-4}

To simplify the rational expression by subtracting the exponents in the denominator from the exponents in the numerator.

3

Equate the simplified exponent of

x

to the exponent of

x

in the given equivalent expression.

2a - 2 = 6

Equivalent expressions must have identical exponents for corresponding variable bases.

4

Solve the linear equation for

a

.

a = 4

To find the constant value that satisfies the equation.

5

Verify the value of

a

using the exponents of

y

.

3(4) - 4 = 8

, which is true.

To ensure consistency across both variable exponents.

Anahtar Kavram

Equivalent algebraic expressions involving exponent rules

Soru 198Soru →

A parabola and a line intersect at exactly one point in the $xy$ -plane. The parabola is defined by the equation $y = -x^2 + 6x - 2$ and the line is defined by the equation $y = 2x + k$ , where $k$ is a constant. What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

To find the value of

k

where the parabola and the line intersect at exactly one point, we equate the two equations:

-x^2 + 6x - 2 = 2x + k

. Rearranging this into standard quadratic form yields

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

. For a quadratic equation to have exactly one real solution, its discriminant,

b^2 - 4ac

, must be equal to zero. Substituting

a = 1

,

b = -4

, and

c = k + 2

into the discriminant formula gives

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

, which simplifies to

16 - 4k - 8 = 0

, or

8 - 4k = 0

. Solving for

k

gives

k = 2

.

Adım Adım Çözüm

1

Equate the equations of the parabola and the line to set up an equation for the x-coordinates of their intersection points.

-x^2 + 6x - 2 = 2x + k

At the points of intersection, the y-values of both equations are equal.

2

Rearrange the equation into the standard quadratic form

ax^2 + bx + c = 0

.

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

This allows us to identify the coefficients

a = 1

,

b = -4

, and

c = k + 2

to apply the quadratic discriminant.

3

Set the discriminant

b^2 - 4ac

equal to zero.

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

A quadratic system has exactly one real solution (tangency) if and only if the discriminant of the resulting quadratic equation is zero.

4

Solve the linear equation for

k

.

k = 2

Simplifying the expression yields

16 - 4k - 8 = 0

, which simplifies to

8 - 4k = 0

, giving

k = 2

.

Anahtar Kavram

Nonlinear Systems of Equations

Soru 199Soru →

During a chemical reaction, the temperature $T$ , in degrees Celsius, of a solution $s$ seconds after the reaction begins is modeled by the equation $T = -0.04(s - 150) + 92$ , where $150 \le s \le 900$ . According to the model, how many minutes does it take for the temperature of the solution to decrease by $12$ degrees Celsius?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

To find the number of minutes it takes for the temperature to decrease by

12^\circ\text{C}

, we first determine the rate of temperature change from the linear model. The equation is given in the form

T = m(s - s_0) + T_0

, where the slope

m = -0.04

represents the rate of change of temperature in degrees Celsius per second. Thus, the temperature decreases at a rate of

0.04^\circ\text{C}

per second. To achieve a total decrease of

12^\circ\text{C}

, the time in seconds required is

\frac{12}{0.04} = 300

seconds. Converting

300

seconds to minutes gives

\frac{300}{60} = 5

minutes.

Adım Adım Çözüm

1

Identify the rate of change from the linear equation.

The rate of temperature decrease is

0.04^\circ\text{C}

per second.

The slope of the linear equation

T = -0.04(s - 150) + 92

is

-0.04

, which represents a change of

-0.04^\circ\text{C}

for every

1

second increase in time.

2

Calculate the time in seconds for a decrease of

12^\circ\text{C}

.

300

seconds

Divide the target temperature change of

-12^\circ\text{C}

by the rate of change of

-0.04^\circ\text{C}

per second:

\frac{-12}{-0.04} = 300

seconds.

3

Convert the time from seconds to minutes.

5

minutes

Since there are

60

seconds in

1

minute, divide

300

seconds by

60

:

\frac{300}{60} = 5

minutes.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 200Soru →

A nutritionist is designing a meal plan containing $x$ grams of protein and $y$ grams of carbohydrates. The meal plan must satisfy the following system of inequalities:

\begin{aligned} y &\ge 1.5x + 15 \\ 3x + 2y &\le 120 \end{aligned}

What is the maximum possible number of grams of protein, $x$ , that can be included in the meal plan?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The maximum possible number of grams of protein that can be included is 15.

To find the maximum possible value of

x

, we determine the region defined by the system of inequalities. The system restricts the values to the region above the line

y = 1.5x + 15

and below the line

3x + 2y = 120

. Since the first inequality limits

y

from below and the second limits

y

from above, the feasible region narrows as

x

increases, terminating at the intersection of the two boundary lines. Substituting

y = 1.5x + 15

into

3x + 2y = 120

gives

3x + 2(1.5x + 15) = 120

. Simplifying this yields

3x + 3x + 30 = 120

, which simplifies further to

6x = 90

, giving

x = 15

. Thus, the maximum value of

x

is 15.

Adım Adım Çözüm

1

Identify the boundary lines of the system of inequalities.

The boundary lines are

y = 1.5x + 15

and

3x + 2y = 120

.

The maximum value of

x

under these linear constraints occurs at the intersection of the boundary lines of the feasible region.

2

Substitute the expression for

y

from the first boundary equation into the second equation.

3x + 2(1.5x + 15) = 120

This allows us to solve for

x

by eliminating

y

.

3

Simplify the equation and solve for

x

.

3x + 3x + 30 = 120 \implies 6x + 30 = 120 \implies 6x = 90 \implies x = 15

.

Solving the linear equation gives the

x

-coordinate of the intersection point.

4

Verify that this point lies in the feasible region and represents the maximum possible value of

x

.

At

x=15

,

y=37.5

. Since

y \ge 1.5x + 15

restricts the region above the line and

3x + 2y \le 120

restricts it below the line, the region lies to the left of the intersection point

(15, 37.5)

. Thus, the maximum value of

x

is 15.

Confirming the geometry of the feasible region ensures the intersection point is indeed the maximum value.

Anahtar Kavram

Solving systems of linear inequalities to find the boundaries and extreme values of a feasible region.

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