Set 9

Explanation

Given the area of the large square is $256 \text{ cm}^2$.

Finding the Side Length of the Large Square

The area of a square ($A$) is given by $A = s^2$.

Finding the Diameter of the Circle

Notice that the circle is inscribed in the large square, tangent to its sides. Therefore, the diameter of the circle ($d$) is equal to the side length of the large square.

Calculating the Circumference of the Circle

The circumference of a circle ($C$) is given by $C = \pi d$.

Therefore, the circumference of the circle is $16\pi \text{ cm}$.

Explanation

Let $F$ be the center of the circle, so $AF = FD = FG = r$ (the radius of the circle).

Given the side length of the square $AD = 24 \text{ cm}$. Since $E$ is the midpoint of $AD$ (based on the symmetry of the circle passing through $A$ and $D$), then:

Finding the Radius of the Circle

We will find the radius of the circle ($r$) using the Pythagorean theorem on the right-angled triangle $\triangle AEF$. Note that the length $EF$ is the distance from the center of the circle to side $AD$. Since the circle is tangent to side $BC$ at point $G$, and the total distance from side $AD$ to side $BC$ is the side length of the square ($24 \text{ cm}$), then $EF = AB - r = 24 - r$.

Using the Pythagorean theorem on $\triangle AEF$:

Calculating the Area of the Circle

After finding the radius $r = 15 \text{ cm}$, we can calculate the area:

Therefore, the area of the circle is $225\pi \text{ cm}^2$.

Explanation

We draw an auxiliary line parallel to lines $PQ$ and $RS$ passing through the vertex of angle $\alpha$. This line divides angle $\alpha$ into two parts, $\beta$ and $\gamma$.

Finding the Value of $\beta$

Notice that angle $\beta$ and the $25^\circ$ angle are alternate interior angles. Since the lines are parallel, their measures are equal:

Finding the Value of $\gamma$

Angle $\gamma$ and the $135^\circ$ angle are consecutive interior angles. The sum of these two angles is $180^\circ$:

Calculating the Value of $\alpha$

The value of $\alpha$ is the sum of $\beta$ and $\gamma$:

Therefore, the value of $\alpha$ is $70^\circ$.

Explanation

Let $n_A$ be the number of students in class A and $n_B$ be the number of students in class B.

Given the ratio of the average scores of class A and class B is:

And the ratio of the combined average to the average of class $A$ is:

The ratio of the number of students in class A to class B can be found using the combined average formula:

Substitute the values of $\bar{X}$ and $\bar{X}_B$ in terms of $\bar{X}_A$:

Divide both sides by $\bar{X}_A$ (since $\bar{X}_A \neq 0$):

Cross-multiply:

Group like terms:

Simplify the equation to find the ratio $n_A : n_B$:

Therefore, the ratio of the number of students in class A to class B is 2 : 1.

Explanation

Let $A$ be the event that the number of green balls drawn is twice the number of white balls drawn.

Given:

There are $9$ balls in a box ($1$ red + $6$ green + $2$ white = $9$). $7$ balls will be drawn at random. The total number of ways to draw the balls is:

We want to find the probability that the number of green balls is twice the number of white balls. We divide this into two cases.

Case 1

If $1$ white ball is drawn, then there must be $2$ green balls (since green = 2 times white). To reach a total of $7$ balls drawn, the remaining balls to be drawn must be $7 - 1 - 2 = 4$ balls. These must be red balls. However, there is only $1$ red ball available. Therefore, Case 1 is impossible.

Case $2$

If $2$ white balls are drawn, then there must be $4$ green balls. To reach a total of $7$ balls drawn, the remaining balls to be drawn must be $7 - 2 - 4 = 1$ ball. This must be a red ball. This case is possible because there are $2$ white balls, $6$ green balls, and $1$ red ball available.

The number of ways to draw for this case is:

Conclusion

Thus, the probability of event $A$ is:

Therefore, the probability that the number of green balls drawn is twice the number of white balls drawn is $\frac{5}{12}$.

Explanation

Given $1 < x < y < 2$. We want to find the range of values for $z = x - y$.

Since $x < y$, the difference $x - y$ must be negative ($z < 0$).

Let's analyze the limits of $z$:

1. Maximum value of $z$

   The value of $z$ will be maximum (closest to $0$) when the difference between $x$ and $y$ is as small as possible.

   Since $x$ and $y$ are in the interval $(1, 2)$ and $x < y$, their difference can be arbitrarily close to $0$ but never reaches $0$ (because $x \neq y$).

   Thus, the upper bound of $z$ is $0$.

2. Minimum value of $z$

   The value of $z$ will be minimum (most negative) when $x$ is as small as possible and $y$ is as large as possible.

   • $x$ approaches $1$ (lower bound).
   • $y$ approaches $2$ (upper bound).

   Therefore:

   $$z \approx 1 - 2 = -1$$

   Thus, the lower bound of $z$ is $-1$.

Conclusion

The value of $z$ is between $-1$ and $0$.

In mathematical notation: $-1 < z < 0$.

Therefore, $z$ is between the values $-1$ and $0$.

Explanation

Let:

• $a =$ current age of the younger sibling
• $k =$ current age of the older sibling

First Condition Analysis

Seven years ago, the sum of the ages of the older sibling and the younger sibling was $26$ years.

The mathematical model is:

Second Condition Analysis

Four years from now, twice the age of the older sibling is equal to twice the age of the younger sibling plus eight years.

The mathematical model is:

Finding the Younger Sibling's Age

We eliminate equation $(1)$ and $(2)$ to find the value of $a$.

We subtract the second equation from the first: $(k+a) - (k-a) = 2a$ and $40 - 4 = 36$.

Thus, the current age of the younger sibling is $18$ years.

Therefore, the younger sibling is $18$ years old.

Explanation

Simplifying Algebraic Expression

We simplify the numerator by grouping like terms (integers with integers, radicals with radicals).

Next, we factor the numerator by taking out the number $3$:

Notice that $(3 + \sqrt{5})$ is equal to $(\sqrt{5} + 3)$. Thus, we can divide both the numerator and the denominator by this value:

Therefore, the final result is $3$.

Explanation

To find the domain of the function, we need to ensure that all components of the function are defined (values inside square roots must be non-negative, and the denominator cannot be zero).

The function $f(x) = \frac{\sqrt{5 - x}}{\sqrt{x^2 + x - 6}}$ has two main conditions:

Condition 1: Numerator

The value inside the square root in the numerator must be non-negative:

Condition 2: Denominator

The value inside the square root in the denominator must be positive (since the square root in the denominator cannot be zero):

The zeros of this inequality are $x = -3$ and $x = 2$.

Using test points, we find the solution region for the denominator: $x < -3$ or $x > 2$.

Domain Intersection

The domain of function $f$ is the intersection of the two conditions above:

1. $x \leq 5$
2. $x < -3$ or $x > 2$

The intersection is:

• For $x < -3$, it satisfies $x \leq 5$.
• For $x > 2$, it is bounded by $x \leq 5$, becoming $2 < x \leq 5$.

Therefore, the domain of function $f$ is:

Explanation

Finding the Side Length Ratio

The volume of a cube ($V$) is directly proportional to the cube of its side length ($s$). Given that the volume ratio is $1 : 27$.

So, the ratio of the side lengths of the two cubes is $1 : 3$.

Finding the Surface Area Ratio

The surface area of a cube ($L$) is directly proportional to the square of its side length.

Thus, the ratio of the surface areas of the two cubes is $1 : 9$.

Explanation

Let:

• $P(A)$ = Probability Ali passes
• $P(B)$ = Probability Bima passes
• $P(D)$ = Probability Dika passes

Given:

If only one person passes, there are three mutually exclusive possibilities:

Ali passes, Bima and Dika fail

2. Bima passes, Ali and Dika fail

Dika passes, Ali and Bima fail

Total Probability

Thus, the probability that only one person passes is the sum of these three possibilities:

Explanation

This problem can be solved using inverse proportion.

Given:

Calculating the Number of Workers Needed

Thus, we can write:

So, the number of workers needed to finish in $9$ days is $20$ workers.

Calculating the Percentage Increase

Thus, the percentage increase in workers is:

Explanation

Properties of Average and Range

• Average: Its value changes if each data point is added, subtracted, multiplied, or divided.
• Range: Its value only changes if each data point is multiplied or divided (it does not change if added or subtracted).

Finding the Value of $b$

The initial range is $12$. Then, each data value is subtracted by $a$ and divided by $b$. Since the range only changes when multiplied or divided, the new range becomes:

Finding the Value of $a$

The initial average is $24$. Then, each data value is subtracted by $a$ and divided by $b$, so the new average becomes:

Substitute the value $b = 2$:

Thus, the values are $a = 4$ and $b = 2$.

Explanation

Converting to Fractions

To determine the order from smallest to largest, we convert all numbers into fractions with a common denominator.

Finding a Common Denominator

We use $2100$ as the common denominator (LCM of $100$, $14$, and $3$).

For $\sqrt{12}$, since its value is approximately $3.46$, if converted to a fraction over 2100, it would be very large (much larger than the others which are all below 2).

Conclusion

Based on the numerators with the common denominator:

So the order is:

Returning to the original forms:

Explanation

Calculating the Value of $x$

Given $x = \frac{3}{4}$ of $34$, which means:

Calculating the Value of $y$

Given $y = 46\%$ of $86$, then:

Conclusion

Based on the calculations above, we get $x = 25.5$ and $y = 39.56$.

Consequently, $x < y$.

Explanation

Let $x = 3.15931593...$.

Since there are $4$ repeating digits (which are $1593$), we multiply $x$ by $10000$:

Next, we subtract the original equation $x$ from this equation:

Thus, we obtain:

We simplify the fraction by dividing both the numerator and the denominator by $9$:

Explanation

Finding the Relation between $p$ and $l$

Given that the perimeter of the rectangle is $6l$. The formula for the perimeter of a rectangle is $2(p + l)$ or $2p + 2l$. Thus:

Finding the Value of $l$

Next, it is known that the area of the rectangle is $32$. The formula for the area is $p \times l$. We substitute $p = 2l$ into the area formula:

Calculating the Value of $p$

After obtaining the value $l = 4$, we substitute it back into the equation $p = 2l$: