Set 3

Explanation

Delivery Period Analysis

April $2023$ has a total of $30$ days. The first delivery was made on April $1$, which corresponds to day $1$. We need to calculate how many times suppliers A and B delivered goods within that $30$-day period.

The delivery schedule follows an Arithmetic Progression pattern with the first term $a = 1$ (first delivery date) and common difference $b$ corresponding to each supplier's interval. The condition for the delivery date is $U_n \le 30$.

Cooking Oil A Volume

Supplier A delivers every $3$ days.

Thus, supplier A delivers $10$ times.

Total volume A:

Cooking Oil B Volume

Supplier B delivers every $4$ days.

Thus, supplier B delivers $8$ times.

Total volume B:

Combined Total Volume

Therefore, the total volume of cooking oil brands A and B received is $220$ liters.

Explanation

Calculating Net Weight (Netto)

We need to determine the actual amount of rice available for sale. Netto is the gross weight minus the packaging weight (tare).

Calculating Loss

The merchant incurs a loss because the total selling price is lower than the purchase price.

Therefore, the merchant incurs a loss of $\text{Rp}155,000.00$.

Explanation

Net Weight Analysis (Netto)

First, we calculate how much watermelon is actually sold by subtracting the tare (packaging weight) from the gross weight (bruto).

Total Revenue (Selling Price)

Next, we calculate the total revenue from the watermelon sales. The remaining weight of the watermelons is $95 - 60 - 20 = 15 \text{ kg}$.

Calculating Purchase Price

The merchant makes a $70\%$ profit, meaning the Selling Price is $170\%$ of the Purchase Price.

Thus, the purchase price for one crate of watermelons is $\text{Rp}250,000.00$.

Explanation

Identifying Geometric Relationships

It is given that the radius ($r$) of the package is half of its height ($t$), so we can write $r = \frac{1}{2}t$. The volume of the cylindrical package is given as $V = \frac{729\pi}{4} \text{ cm}^3$.

Calculating the Height

Using the cylinder volume formula, we substitute the value of $r$ to solve for $t$.

Thus, the height of the package is $9 \text{ cm}$.

Explanation

Formula Analysis

The lateral surface of a cylinder is a rectangle with length $2\pi r$ and height $t$. Thus, the formula for the lateral surface area is $L_S = 2\pi r t$.

We will compare the volume ($V = \pi r^2 t$) to the lateral surface area ($L_S$).

Calculating the Ratio

In the previous question (Number 4), it was determined that the package height $t = 9 \text{ cm}$. Since $r = \frac{1}{2}t$, then $r = \frac{9}{2}$.

Substitute the value of $r$ into the ratio equation:

Thus, the ratio is $9:4$.

Explanation

First Offer Analysis

Supri receives a fixed salary of $\text{Rp}75,000.00$ per day for $7$ days.

Second Offer Analysis

Supri receives $\text{Rp}10,000.00$ on the first day, doubling each subsequent day. This is a geometric series with $a = 10,000$, $r = 2$, and $n = 7$.

Conclusion

Comparing the total income from both offers:

Since the total income from the second offer is significantly higher, the best choice is the Second offer.

Explanation

Forming Equations

The profit increases by a constant amount, so this is an arithmetic progression. The total profit up to the $n$-th day is denoted as $S_n$.

Given:

1. Total profit up to the $4$th day ($S_4$) is $\text{Rp}700,000.00$.
2. Total profit up to the $10$th day ($S_{10}$) is $\text{Rp}2,200,000.00$.

Formula for the sum of $n$ terms:

From $S_4 = 700,000$:

From $S_{10} = 2,200,000$:

Elimination and Substitution

Eliminating equations $(1)$ and $(2)$:

Substituting $b = 15,000$ into equation $(1)$:

Calculating Total Profit for Day $20$

We calculate $S_{20}$:

Thus, the total profit up to the $20$th day is $\text{Rp}5,900,000.00$.

Explanation

Concept of Break-Even Point

The Break-Even Point is the condition where there is neither profit nor loss, meaning total revenue equals total cost.

Solving the Equation

Substitute the given functions:

To make calculation easier, multiply both sides by $10$:

Determining the Value of x

We use quadratic factorization. We look for two numbers that multiply to $3600$ and add up to $-200$. These numbers are $-180$ and $-20$.

The roots are:

Thus, the break-even point occurs when the number of products is $180$ units and $20$ units.

Explanation

Equating Inflow and Outflow Volumes

The required condition is when the volume of water entering equals the volume of water flowing out.

Substitute the known functions:

Solving the Quadratic Equation

Move all terms to the left side to get the standard quadratic form:

Simplify the equation by dividing both sides by $2$:

Factor the equation. We look for two numbers that multiply to $2100$ and add up to $-100$. These numbers are $-70$ and $-30$.

This gives the solutions:

Thus, the volume of entering water equals the outflowing water at $70$ minutes or $30$ minutes.

Explanation

Calculating Total Balls

The bag contains $4$ green balls and $5$ black balls.

Probability of First Green Ball

The probability of drawing a green ball on the first draw is the number of green balls divided by the total number of balls.

Probability of Second Green Ball

Since the draw is without replacement, the number of green balls decreases to $3$ and the total number of balls decreases to $8$.

Calculating Total Probability

The probability that both balls are green is the product of these two probabilities:

Thus, the probability that both balls are green is $\frac{1}{6}$.

Explanation

Filling Slots Method

We have $7$ flags with a specific condition: Armenia and Belgium must be at the ends of the row. We can visualize the arrangement using slots (boxes).

Step 1: Determine End Positions

There are only $2$ possibilities for the ends: Armenia on the left and Belgium on the right, or vice versa. Thus, we write the number $2$ in the first box as the multiplier for the end variations.

Step 2: Fill Middle Positions

There are $5$ remaining flags (China, Denmark, Ecuador, Finland, Ghana) to fill the $5$ middle positions. We fill the remaining slots in descending order ($5, 4, 3, 2, 1$) following the permutation principle.

Here is the visualization of the filled slots:

The number $2$ represents the ways to arrange the ends.

The numbers $5, 4, 3, 2, 1$ represent the ways to arrange the $5$ middle flags ($5!$).

Final Calculation

The total number of ways is the product of all numbers in the boxes.

Thus, there are $2(5!)$ ways to arrange the flags.

Explanation

Ratio Analysis

The ratio of visitors on Monday, Tuesday, and Wednesday is given as $3 : 4 : 5$. The total number of visitors for these three days is $144$.

We can calculate the total number of parts (units) in the ratio:

This means $12$ units correspond to $144$ people.

Calculating Monday's Visitors

Monday has a share of $3$ units. We use the fraction method to find the number of visitors.

Simplify the fraction $\frac{3}{12}$ to $\frac{1}{4}$.

Thus, the number of visitors on Monday is $36$.

Explanation

Proportion Analysis

We have the visitor ratio for Thursday, Friday, and Saturday as follows:

It is known that Friday accounts for $x$ parts of this total ratio. The number of Friday visitors is $24$ out of a total of $120$ people.

Calculating the Value of x

We can set up an equation using the fractional part concept:

Simplify the equation by dividing both sides by $24$ (since $120 \div 24 = 5$):

Thus, the value of $x$ is $2$.

Explanation

Mathematical Model

Let the money of Amri, Beni, and Dimas be $A, B,$ and $D$ respectively. Based on the problem, we can form a system of equations:

Solving the System

We want to find the value of $A$. Substitute equation (1) into equation (2):

Next, we eliminate $B$ using equations (3) and (4). Multiply equation (3) by $2$ so the coefficients of $B$ match:

Subtract the two equations:

Substitute $D = 14,000$ into equation (3) to find $B$:

Finally, substitute the values of $B$ and $D$ into equation (1) to find $A$:

Thus, Amri's money is $\text{Rp}67,000.00$.

Explanation

Work Rate Analysis

Work rate is the portion of work completed per unit of time (1 minute). Based on the problem:

Combined Work Calculation

If the three machines work together for $6$ minutes, the total work completed is the sum of their rates multiplied by time:

Simplify the fraction $\frac{4}{32}$ to $\frac{1}{8}$:

In decimal form, $\frac{3}{4}$ is equal to $0.75$.

Thus, the completed work is $0.75$ part.

Explanation

We define the variables based on the given information:

• $n_A = 20$ (Number of food agents)
• $n_B = 40$ (Number of beverage agents)
• $\bar{x}_A$ (Average profit of food agents)
• $\bar{x}_B$ (Average profit of beverage agents)

It is known that the average profit of food agents is twice that of beverage agents, so we have the relationship:

We substitute this relationship into the total profit equation to find the value of $\bar{x}_A$ directly:

Thus, the average profit of food agents is:

Therefore, the average profit of agents selling food products is Rp9,000,000.00.

Explanation

Let the four sorted natural numbers be $x_1, x_2, x_3, x_4$ where $x_1 \le x_2 \le x_3 \le x_4$.

Based on the problem information, we can form the following equations:

1. Median: Since there are an even number of data points (4), the median is the average of the two middle values.

   $$\frac{x_2 + x_3}{2} = 8 \implies x_2 + x_3 = 16$$

2. Range: The difference between the largest and smallest datum is 10.

   $$x_4 - x_1 = 10 \implies x_1 = x_4 - 10$$

3. Mean: The sum of all data divided by 4 is 8.

   $$\frac{x_1 + x_2 + x_3 + x_4}{4} = 8 \implies x_1 + x_2 + x_3 + x_4 = 32$$

We substitute the value $x_2 + x_3 = 16$ into the mean equation to find the relationship between $x_1$ and $x_4$:

Now we have a system of two equations for $x_1$ and $x_4$:

By adding both equations, we get:

Substituting the value of $x_4$ to get $x_1$:

Finally, we determine the values of $x_2$ and $x_3$. We know $x_2 + x_3 = 16$ and $3 \le x_2 \le x_3 \le 13$. Since the mode is unique, there must be one value that appears more than once. The only possibility satisfying the median condition and the order is if $x_2$ and $x_3$ are equal.

The complete data set is: $3, 8, 8, 13$. The mode of this data is $8$ (unique), which satisfies the problem condition.

Thus, the product of the second and fourth datum is:

Explanation

We will find the height of the helicopter from the helipad ($DC$) using trigonometric ratios in right-angled triangles.

Observe the solution sketch:

Observe the geometry:

• $AD = 16$ m (building height).
• $AB$ is the distance from the building to the tip of the shadow.
• The angle of elevation to the top of the building is $30^\circ$ and to the helicopter is $45^\circ$.

First, we calculate the distance $AB$ using $\tan(30^\circ)$ in triangle $ABD$.

Next, use this distance $AB$ to find the total height of the helicopter from the ground ($AC$) using $\tan(45^\circ)$ in triangle $ABC$.