Quiz: Foundations: Algebra

What is the solution to the equation? $3x - 5 = 16$

Select the correct answer

$x = 3$

$x = 5$

$x = 7$

$x = 9$

Explanation

3x - 5 = 16

Add 5 to both sides of the equation:
$3x = 16 + 5$ $3x = 21$
Divide both sides by 3:
$x = \frac{21}{3} = 7$

Quiz: Foundations: Problem solving and data analysis

Olga went to a store and bought a sweater originally priced at 1200 USD. The store had a promotion:

First, a 15% discount was applied to the sweater.
Then, an additional 10% discount was applied to the remaining amount.

What is the final price of the sweater after all the discounts?

Select the correct answer

918 USD

1020 USD

1100 USD

920 USD

Explanation

First discount (15% off 1200 USD):
$1200 \times 0.15 = 180$
New price after the first discount:
$1200 - 180 = 1020$
Second discount (10% off 1020 USD):
$1020 \times 0.10 = 102$
Final price:
$1020 - 102 = 918$

Quiz: Foundations: Advanced math

$f(n) = 400 - 20n + k$

In the function above, k is a constant. If f(5) = 310, what is the value of f(7)?

Select the correct answer

$250$

$270$

$290$

$300$

Explanation

We start with the given equation:

f(n) = 400 - 20n + k

Step 1: Find $k$

We are given that $f(5) = 310$ , so we substitute $n = 5$ into the equation:

400 - 20(5) + k = 310

Calculate the term $20(5)$ :

400 - 100 + k = 310

Rearrange to solve for $k$ :

k = 310 - 300

k = 10

Now that we have $k = 10$ , we can use it to find $f(7)$ .

Step 2: Find $f(7)$
Substituting $n = 7$ into the equation:

f(7) = 400 - 20(7) + 10

Calculate $20(7)$ :

f(7) = 400 - 140 + 10 = 270

Thus, the total weight of the cargo after 7 trips is 270 kg.

Quiz: Foundations: Geometry and trigonometry

A baker is making chocolate truffles, which are shaped like perfect spheres. One truffle has a radius of 6 centimeters. Find its volume in $\text{cm}^3$ .

Select the correct answer

$250\pi \, \text{cm}^3$

$270\pi \, \text{cm}^3$

$288\pi \, \text{cm}^3$

$300\pi \, \text{cm}^3$

Explanation

The volume of a sphere is given by the formula:

V = \dfrac{4}{3} \pi r^3

Substituting $r = 6$ cm into the formula:

V = \dfrac{4}{3} \pi (6)^3 = \dfrac{4}{3} \pi (216) = \dfrac{864}{3} \pi = 288\pi \, \text{cm}^3

Thus, the volume of the sphere is $288\pi \text{cm}^3$ .

Quiz: Medium: Algebra

\begin{cases} 3x + 2y = 12 \\ 5x - y = 7 \end{cases}

Which ordered pair (x, y) satisfies the system of equations above?

Select the correct answer

$(2, 3)$

$(1, 4)$

$(-3, 5)$

$(0, -2)$

Explanation

We have the system of equations:

\begin{cases} 3x + 2y = 12 \\ 5x - y = 7 \end{cases}

Step 1: Express $y$ in terms of $x$ from the second equation

y = 5x - 7

Step 2: Substitute this expression into the first equation

3x + 2(5x - 7) = 12

Expanding the parentheses:

3x + 10x - 14 = 12

Simplifying:

13x - 14 = 12

13x = 26

x = 2

Step 3: Find $y$ by substituting $x = 2$ into $y = 5x - 7$

y = 5(2) - 7 = 10 - 7 = 3

Quiz: Medium: Problem solving and data analysis

A farmer has 360 hectares of wheat fields, and the ratio of field area to the number of tractors needed for cultivation is 30:4. How many tractors does the farmer need to cultivate the entire field?

Write your answer

Quiz: Medium: Advanced math

Find the sum of the expressions:

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

Select the correct answer

$x+1$

$x+3$

$x+4$

$x-1$

Explanation

Since both fractions have the same denominator, we add the numerators and keep the denominator:

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

Factorizing the numerator:

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

Canceling the common factor $x+3$ :

x+1

Quiz: Medium: Geometry and trigonometry

Archaeologists discovered an ancient amulet in the shape of a regular polygon with n sides. After measurements, they found that the average interior angle of this polygon is 165°. How many sides does this polygon have?

Select the correct answer

Explanation

In a regular n-sided polygon, each interior angle is given by the formula:

\text{Interior angle} = \frac{180(n-2)}{n}

where n is the number of sides in the polygon.

Substituting the given angle:

\frac{180(n-2)}{n} = 165

180(n-2) = 165n

180n - 360 = 165n

180n - 165n = 360

15n = 360

n = 24

Quiz: Advanced: Algebra

Given the parametric equations:

\begin{aligned} x &= t^2 + t,\\ y &= t^2 - 2t. \end{aligned}

Determine which of the following statements correctly describes the possible range of values for $y$ on the graph of these equations.

Select the correct answer

$y \geq 0$

$y \geq -\dfrac{1}{4}$

$y \geq -1$

$y \leq 1$

Explanation

Consider the equation for $y$ :

y = t^2 - 2t.

To find the minimum value of $y$ , we use the method of completing the square. Rewrite the expression as follows:

y = t^2 - 2t.

Add and subtract 1:

y = (t^2 - 2t + 1) - 1.

Rewriting:

y = (t - 1)^2 - 1.

Since the square of any number is always non-negative:

(t - 1)^2 \geq 0,

we obtain:

y = (t - 1)^2 - 1 \geq -1.

Thus, $y$ can take values no less than $-1$ , but it has no upper limit.

Quiz: Advanced: Problem solving and data analysis

At a bridge construction site, the ratio of workers to engineers is 36 to 5.

Additionally, the ratio of engineers to managers is 10 to 3.

What is the ratio of managers to workers?

Select the correct answer

1 to 24

3 to 50

5 to 42

3 to 60

Explanation

We are given two ratios:

Workers to Engineers:
$36:5$
This means that for every 36 workers, there are 5 engineers.
Engineers to Managers:
$10:3$
This means that for every 10 engineers, there are 3 managers.

Finding the ratio of managers to workers

To merge these ratios, we need to equalize the number of engineers. In the first ratio, there are 5 engineers, and in the second, there are 10 engineers.

Multiply the first ratio by 2 to make the number of engineers equal to 10:

(36 \times 2) : (5 \times 2) = 72:10

Now we know that 10 engineers correspond to 72 workers.

From the second ratio, we know that 10 engineers correspond to 3 managers.

Thus, the complete ratio is:

72 \text{ workers} : 10 \text{ engineers} : 3 \text{ managers}

From this, the ratio of managers to workers is:

3:72

Dividing both numbers by 3:

1:24

Quiz: Advanced: Advanced math

Let the given matrix be

\begin{bmatrix} 4p & 2 \\ -5 & 3 \end{bmatrix}.

Find its determinant.

Select the correct answer

$12p - 10$

$12p + 10$

$10p + 12$

$10p - 12$

Explanation

To find the determinant of a $2 \times 2$ matrix, we use the formula:

\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}.

Now, calculate the determinant:

\text{det}(A) = (4p) \times 3 - 2 \times (-5) = 12p + 10.

Quiz: Advanced: Geometry and trigonometry

A sphere with a diameter of 10 meters is inscribed in a cube. Find the volume of the space between the sphere and the cube.

Select the correct answer

450 m³

476.4 m³

523.6 m³

500 m³

Explanation

Calculate the volume of the sphere:
- Radius of the sphere: $r = \frac{10}{2} = 5 \text{ m}$
- Volume formula for a sphere: $V{\text{sphere}} = \frac{4}{3} \pi r^3$
- Substituting the values: $V{\text{sphere}} = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \approx 523.6 \text{ m}^3$
Calculate the volume of the cube:
- The side length of the cube is equal to the diameter of the sphere, which is 10 m. $V{\text{cube}} = a^3 = 10^3 = 1000 \text{ m}^3$
Calculate the volume of the space between the cube and the sphere:

V{\text{space}} = V{\text{cube}} - V{\text{sphere}}

V{\text{space}} = 1000 - 523.6 = 476.4 \text{ m}^3

Quiz: Foundations: Algebra

What is the solution to the equation? $3x - 5 = 16$

Select the correct answer

$x = 3$

$x = 5$

$x = 7$

$x = 9$

Explanation

3x - 5 = 16

Add 5 to both sides of the equation:
$3x = 16 + 5$ $3x = 21$
Divide both sides by 3:
$x = \frac{21}{3} = 7$

Quiz: Foundations: Problem solving and data analysis

Olga went to a store and bought a sweater originally priced at 1200 USD. The store had a promotion:

First, a 15% discount was applied to the sweater.
Then, an additional 10% discount was applied to the remaining amount.

What is the final price of the sweater after all the discounts?

Select the correct answer

918 USD

1020 USD

1100 USD

920 USD

Explanation

First discount (15% off 1200 USD):
$1200 \times 0.15 = 180$
New price after the first discount:
$1200 - 180 = 1020$
Second discount (10% off 1020 USD):
$1020 \times 0.10 = 102$
Final price:
$1020 - 102 = 918$

Quiz: Foundations: Advanced math

$f(n) = 400 - 20n + k$

In the function above, k is a constant. If f(5) = 310, what is the value of f(7)?

Select the correct answer

$250$

$270$

$290$

$300$

Explanation

We start with the given equation:

f(n) = 400 - 20n + k

Step 1: Find $k$

We are given that $f(5) = 310$ , so we substitute $n = 5$ into the equation:

400 - 20(5) + k = 310

Calculate the term $20(5)$ :

400 - 100 + k = 310

Rearrange to solve for $k$ :

k = 310 - 300

k = 10

Now that we have $k = 10$ , we can use it to find $f(7)$ .

Step 2: Find $f(7)$
Substituting $n = 7$ into the equation:

f(7) = 400 - 20(7) + 10

Calculate $20(7)$ :

f(7) = 400 - 140 + 10 = 270

Thus, the total weight of the cargo after 7 trips is 270 kg.

Quiz: Foundations: Geometry and trigonometry

A baker is making chocolate truffles, which are shaped like perfect spheres. One truffle has a radius of 6 centimeters. Find its volume in $\text{cm}^3$ .

Select the correct answer

$250\pi \, \text{cm}^3$

$270\pi \, \text{cm}^3$

$288\pi \, \text{cm}^3$

$300\pi \, \text{cm}^3$

Explanation

The volume of a sphere is given by the formula:

V = \dfrac{4}{3} \pi r^3

Substituting $r = 6$ cm into the formula:

V = \dfrac{4}{3} \pi (6)^3 = \dfrac{4}{3} \pi (216) = \dfrac{864}{3} \pi = 288\pi \, \text{cm}^3

Thus, the volume of the sphere is $288\pi \text{cm}^3$ .

Quiz: Medium: Algebra

\begin{cases} 3x + 2y = 12 \\ 5x - y = 7 \end{cases}

Which ordered pair (x, y) satisfies the system of equations above?

Select the correct answer

$(2, 3)$

$(1, 4)$

$(-3, 5)$

$(0, -2)$

Explanation

We have the system of equations:

\begin{cases} 3x + 2y = 12 \\ 5x - y = 7 \end{cases}

Step 1: Express $y$ in terms of $x$ from the second equation

y = 5x - 7

Step 2: Substitute this expression into the first equation

3x + 2(5x - 7) = 12

Expanding the parentheses:

3x + 10x - 14 = 12

Simplifying:

13x - 14 = 12

13x = 26

x = 2

Step 3: Find $y$ by substituting $x = 2$ into $y = 5x - 7$

y = 5(2) - 7 = 10 - 7 = 3

Quiz: Medium: Problem solving and data analysis

A farmer has 360 hectares of wheat fields, and the ratio of field area to the number of tractors needed for cultivation is 30:4. How many tractors does the farmer need to cultivate the entire field?

Write your answer

Quiz: Medium: Advanced math

Find the sum of the expressions:

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

Select the correct answer

$x+1$

$x+3$

$x+4$

$x-1$

Explanation

Since both fractions have the same denominator, we add the numerators and keep the denominator:

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

Factorizing the numerator:

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

Canceling the common factor $x+3$ :

x+1

Quiz: Medium: Geometry and trigonometry

Select the correct answer

Explanation

In a regular n-sided polygon, each interior angle is given by the formula:

\text{Interior angle} = \frac{180(n-2)}{n}

where n is the number of sides in the polygon.

Substituting the given angle:

\frac{180(n-2)}{n} = 165

180(n-2) = 165n

180n - 360 = 165n

180n - 165n = 360

15n = 360

n = 24

Quiz: Advanced: Algebra

Given the parametric equations:

\begin{aligned} x &= t^2 + t,\\ y &= t^2 - 2t. \end{aligned}

Determine which of the following statements correctly describes the possible range of values for $y$ on the graph of these equations.

Select the correct answer

$y \geq 0$

$y \geq -\dfrac{1}{4}$

$y \geq -1$

$y \leq 1$

Explanation

Consider the equation for $y$ :

y = t^2 - 2t.

To find the minimum value of $y$ , we use the method of completing the square. Rewrite the expression as follows:

y = t^2 - 2t.

Add and subtract 1:

y = (t^2 - 2t + 1) - 1.

Rewriting:

y = (t - 1)^2 - 1.

Since the square of any number is always non-negative:

(t - 1)^2 \geq 0,

we obtain:

y = (t - 1)^2 - 1 \geq -1.

Thus, $y$ can take values no less than $-1$ , but it has no upper limit.

Quiz: Advanced: Problem solving and data analysis

At a bridge construction site, the ratio of workers to engineers is 36 to 5.

Additionally, the ratio of engineers to managers is 10 to 3.

What is the ratio of managers to workers?

Select the correct answer

1 to 24

3 to 50

5 to 42

3 to 60

Explanation

We are given two ratios:

Workers to Engineers:
$36:5$
This means that for every 36 workers, there are 5 engineers.
Engineers to Managers:
$10:3$
This means that for every 10 engineers, there are 3 managers.

Finding the ratio of managers to workers

To merge these ratios, we need to equalize the number of engineers. In the first ratio, there are 5 engineers, and in the second, there are 10 engineers.

Multiply the first ratio by 2 to make the number of engineers equal to 10:

(36 \times 2) : (5 \times 2) = 72:10

Now we know that 10 engineers correspond to 72 workers.

From the second ratio, we know that 10 engineers correspond to 3 managers.

Thus, the complete ratio is:

72 \text{ workers} : 10 \text{ engineers} : 3 \text{ managers}

From this, the ratio of managers to workers is:

3:72

Dividing both numbers by 3:

1:24

Quiz: Advanced: Advanced math

Let the given matrix be

\begin{bmatrix} 4p & 2 \\ -5 & 3 \end{bmatrix}.

Find its determinant.

Select the correct answer

$12p - 10$

$12p + 10$

$10p + 12$

$10p - 12$

Explanation

To find the determinant of a $2 \times 2$ matrix, we use the formula:

\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}.

Now, calculate the determinant:

\text{det}(A) = (4p) \times 3 - 2 \times (-5) = 12p + 10.

Quiz: Advanced: Geometry and trigonometry

A sphere with a diameter of 10 meters is inscribed in a cube. Find the volume of the space between the sphere and the cube.

Select the correct answer

450 m³

476.4 m³

523.6 m³

500 m³

Explanation

Calculate the volume of the sphere:
- Radius of the sphere: $r = \frac{10}{2} = 5 \text{ m}$
- Volume formula for a sphere: $V{\text{sphere}} = \frac{4}{3} \pi r^3$
- Substituting the values: $V{\text{sphere}} = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \approx 523.6 \text{ m}^3$
Calculate the volume of the cube:
- The side length of the cube is equal to the diameter of the sphere, which is 10 m. $V{\text{cube}} = a^3 = 10^3 = 1000 \text{ m}^3$
Calculate the volume of the space between the cube and the sphere:

V{\text{space}} = V{\text{cube}} - V{\text{sphere}}

V{\text{space}} = 1000 - 523.6 = 476.4 \text{ m}^3

Quiz: Foundations: Algebra

Explanation

Quiz: Foundations: Problem solving and data analysis

Explanation

Quiz: Foundations: Advanced math

Explanation

Step 1: Find kkk

Quiz: Foundations: Geometry and trigonometry

Explanation

Quiz: Medium: Algebra

Explanation

Quiz: Medium: Problem solving and data analysis

Quiz: Medium: Advanced math

Explanation

Quiz: Medium: Geometry and trigonometry

Explanation

Quiz: Advanced: Algebra

Explanation

Quiz: Advanced: Problem solving and data analysis

Explanation

Quiz: Advanced: Advanced math

Explanation

Quiz: Advanced: Geometry and trigonometry

Explanation

Quiz: Foundations: Algebra

Explanation

Quiz: Foundations: Problem solving and data analysis

Explanation

Quiz: Foundations: Advanced math

Explanation

Step 1: Find kkk

Quiz: Foundations: Geometry and trigonometry

Explanation

Quiz: Medium: Algebra

Explanation

Quiz: Medium: Problem solving and data analysis

Quiz: Medium: Advanced math

Explanation

Quiz: Medium: Geometry and trigonometry

Explanation

Quiz: Advanced: Algebra

Explanation

Quiz: Advanced: Problem solving and data analysis

Explanation

Quiz: Advanced: Advanced math

Explanation

Quiz: Advanced: Geometry and trigonometry

Explanation

Step 1: Find $k$

Step 1: Find $k$