an aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. the aquarium is filled with water to a depth of 37 cm. a rock with volume $1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. by how many centimeters does the water level rise? express your answer as a decimal to the nearest 100th.

The density function of the continuous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year, is given in Exercise 3.7 on page 92 as f(x) = {x, 0 < x < 1, 2 - x, 1 ≤ x ≤ 2, 0, elsewhere.

To find the average number of hours per year that families run their vacuum cleaners, we must calculate the expected value of X. This is done by integrating the density function of X over the given range:

E(X) = ∫0,2 x * f(x) dx
      = ∫0,1 x2 dx + ∫1,2 (2-x) x dx
      = (1/3) + (-2 + 4 - 2/3)
      = 8/3

Therefore, the average number of hours per year that families run their vacuum cleaners is 8/3, or approximately 2.67 hours.

To find the proportion of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function, we must calculate the cumulative density function of X. This is done by integrating the density function of X over the given range:

F(X) = ∫0,x f(x) dx
      = ∫0,x x dx + ∫x,2 (2-x) dx
      = (1/2)x2 + 2x - 2

Therefore, the proportion of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function is (1/2)x2 + 2x - 2.

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Christian's company charges $0.8 per mile than Samantha's.

In Mathematics and Geometry, the rate of change can be calculated by using the following mathematical equation (formula);

Rate of change, m = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Rate of change, m = (y₂ - y₁)/(x₂ - x₁)

Next, we would determine the rate of change of the value of y with respect to the value of x for Christian's company;

Rate of change, m = (43.50 - 22.50)/(20 - 10)

Rate of change, m = 21/10

Rate of change, m = 2.1

Therefore, the difference between the charges by Christian's company and Samantha's company can be calculated as follows;

Difference in charges = 2.1 - 1.30

Difference in charges = $0.8.

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The Area of Bella's garden as required to be determined in the task content is the difference of the area of the rectangular backyard and the right triangular fish pond.

It follows from the task content that the area of Bella's trapezoidal garden is to be determined from the given information.

Since the garden and the fish pond are from the rectangular backyard; the sum of their areas is equal to the area of the backyard.

Ultimately, the area of the garden is the difference of the area of the rectangular backyard and the right triangular fish pond.

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Answer: 72°

Step-by-step explanation:

To find the interior angle of this shape, use the formula 180(n-2)/n, where n is the amount of sides. Plugging 5 in for the interior angle of a pentagon, you get 180(3)/5, or 108°.

Using the statement that PR intersects QS, we can see that triangle QOR is isosceles (to get this, look at triangle PQR, and note that because it has 2 equal side lengths, and its last length is not equivalent to the other 2 sides, it is isosceles). Solving for angle PRQ, we know one angle is 108°, and the other two are equal. The total angle in a triangle is 180°, so (180°-108°)/2 = 36° (angles QPR and PRQ).

Since the angle of R = 108°, we can find angle PRS as 108° - 36°, or 72°. Since triangles PQR and QRS are similar (share the same angles and side lengths), we can see that angle RQS and RSQ are both 36°.

Since ORS is a triangle, its angle total is 180°. Since we know the angles ORS and OSR (respectively) already as 72° and 36°, we can subtract these angles to find angle ROS. 180°-72°-36° = 72°

The integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 have been written on a chalkboard by Dayna. She then erased the integers from 1 through 6, as well as their multiplicative inverse $\mod{13}$.

We can find the multiplicative inverse of an integer { a modulo 13 } by using the extended Euclidean algorithm.The integers from 1 to 6 are 1, 2, 3, 4, 5, and 6.The multiplicative inverse of : 1 modulo 13 is 1, 2 modulo 13 is 7,  3 modulo 13 is 9, 4 modulo 13 is 10, 5 modulo 13 is 8, and 6 modulo 13 is 11.The only integer that Dayna does not erase is 12.

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The Answer: The first step is Rearranging the Equation

v=9

v=-5

Step-by-step explanation:

STEP

1

:

Rearrange this Absolute Value Equation

Absolute value equalitiy entered

     4|-v+2|-3 = 25

Another term is moved / added to the right hand side.

     4|-v+2| = 28

STEP

2

:

Clear the Absolute Value Bars

Clear the absolute-value bars by splitting the equation into its two cases, one for the Positive case and the other for the Negative case.

The Absolute Value term is 4|-v+2|

For the Negative case we'll use -4(-v+2)

For the Positive case we'll use 4(-v+2)

STEP

3

:

Solve the Negative Case

     -4(-v+2) = 28

    Multiply

     4v-8 = 28

    Rearrange and Add up

     4v = 36

    Divide both sides by 4

     v = 9

STEP

4

:

Solve the Positive Case

     4(-v+2) = 28

    Multiply

     -4v+8 = 28

    Rearrange and Add up

     -4v = 20

    Divide both sides by 4

     -v = 5

    Multiply both sides by (-1)

     v = -5

    Which is the solution for the Positive Case

STEP

5

:

Wrap up the solution

v=9

v=-5

The number 2 is the first even counting number,4 is the second even number, 6 is the third even number, and so forth. the sum of the first 25th even counting numbers is 650.

Since we are given that the first even counting number is 2 and each subsequent even counting number can be obtained by adding 2 to the previous one. so  by using the formula for finding the 25th term of an arithmetic series  which :

an=a+(n-1)d, where the nth term d is a common difference and a is the first term so, the  25th term is 2 + (25-1)*2 = 2 + 48 = 50. Now for finding the sum of the first 25 even counting numbers, we use the formula which is Sn = n /2 * (a1 + an),  where Sn is the sum of the first n terms of the series, a1 is the first term, and an is the nth term. since in this  n=25, a1=2 and an=50, so  after substituting the  values we get  S25 = 25/2 * (2 + 50) = 25/2 * 52 = 650

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Hence, the cost of fencing the garden is ₹9485. Hence, the cost of plowing the garden is ₹756.

Perimeter is the total distance around the outside of a closed two-dimensional shape. It is the sum of the lengths of all the sides of the shape. For example, the perimeter of a rectangle is found by adding the lengths of all its four sides, whereas the perimeter of a circle is found by multiplying the diameter by π (pi). Perimeter is usually expressed in units of length, such as meters, centimeters, feet, or inches.

Here,

The perimeter of the rectangular garden is twice the sum of its length and width. So, the length of the fence needed to enclose the garden is:

2 × (length + width) = 2 × (175 m + 96 m) = 542 m

Therefore, the cost of fencing the garden at 17.50 per meter is:

Cost of fencing = length of fence × cost per meter

= 542 m × 17.50

= 9485

Hence, the cost of fencing the garden is ₹9485.

To find the cost of plowing the garden, we need to first calculate its area, which is given by:

Area = length × width

= 175 m × 96 m

= 16800 m²

Therefore, the cost of plowing the garden at 4.50 paise per square meter is:

Cost of plowing = area of garden × cost per square meter

= 16800 m² × 0.045

= 756

Hence, the cost of plowing the garden is ₹756.

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Answer:

To solve this equation, we will first simplify the left-hand side using the fact that the square root of a number squared is equal to the absolute value of that number.

So, we have:

| x - 4 | = 4 - x

We can now split this equation into two cases, depending on whether x - 4 is positive or negative:

Case 1: x - 4 ≥ 0

In this case, | x - 4 | = x - 4, so we have:

x - 4 = 4 - x

Simplifying this equation, we get:

2x = 8

x = 4

However, we must check this solution to make sure it satisfies the original equation. Plugging x = 4 back into the original equation, we get:

√(4 - 4)^2 = 4 - 4

√0 = 0

So, x = 4 is a valid solution.

Case 2: x - 4 < 0

In this case, | x - 4 | = -(x - 4), so we have:

-(x - 4) = 4 - x

Simplifying this equation, we get:

-2x + 8 = 4

-2x = -4

x = 2

Again, we must check this solution to make sure it satisfies the original equation. Plugging x = 2 back into the original equation, we get:

√(2 - 4)^2 = 4 - 2

√4 = 2

This is a valid solution.

Therefore, the equation has two solutions: x = 4 and x = 2.

Answer: Bro x = 4

Step-by-step explanation:

Answer: 198 families live farther than 1 mile from the square.

Step-by-step explanation:

We know that there are 238 families that live within 1 mile of the village square. To find the number of families that live farther than 1 mile from the square, we can subtract 238 from the total number of families:

436 - 238 = 198

Therefore, 198 families live farther than 1 mile from the square. We can do this subtraction mentally without needing a calculator.

Answer:

No, is not a right triangle

Step-by-step explanation:

If it is a right triangle Pythagoras theorem do apply.

Since the hypotenuse is the side with 16in, sides are 7 and 14 inches

notice

[tex]\sqrt{7^{2} +16^{2} } = \sqrt{245} \neq 16[/tex]

The concentration of hydrogen ions in a liter of the sports drink is approximately 3.98 x 10^(-3) moles per liter.

The pH of a substance is a measure of its acidity or basicity and is defined as the negative logarithm (base 10) of the hydrogen ion concentration [H+]. The formula for pH is given as pH = -log[H+].

To find the concentration of hydrogen ions in a liter of a sports drink that has a pH of 2.4, we can use the formula pH = -log[H+]. Rearranging this formula, we get [H+] = 10^(-pH).

Substituting the given value of pH into this expression, we get [H+] = 10^(-2.4).

It's worth noting that the hydrogen ion concentration is related to the acidity of a solution; the higher the hydrogen ion concentration, the more acidic the solution. The pH scale ranges from 0 (most acidic) to 14 (most basic), with a pH of 7 being neutral. The sports drink in question has a relatively low pH, indicating that it is quite acidic.

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Answer:

AB = 27

Step-by-step explanation:

AC = AB + BC

[tex]{ \rm{57 = 3x + (4x - 6)}} \\ \\ { \rm{57 = 7x - 6}} \\ \\ { \rm{7x = 57 + 6}} \\ \\ { \rm{7x = 63}} \\ \\ { \rm{x = 9}}[/tex]

Therefore;

[tex]{ \rm{AB = 3x = 3 \times 9}} \\ { \boxed{ \rm{AB = 27}}}[/tex]

When evaluating a data set or statistics, the things to consider are sample size and data quality. It's important to consider these things because they provide insight into the validity of the data and the accuracy of the statistics that are being used.

When presented with a dataset or statistics, there are several things to consider.

Here are two examples of what you need to consider when evaluating a dataset or statistics:

1. Sample size: It's important to consider the sample size because small sample sizes are more likely to be biased. For example, a small sample size might be unrepresentative of a larger population. A sample size of 30 is commonly used to distinguish between small and large samples in statistics. Larger sample sizes are often more representative of the population and produce more reliable statistics.

2. Data quality: The quality of the data is also an important consideration. When evaluating statistics, you must ensure that the data is accurate, relevant, and up-to-date. This is important because using incorrect or outdated data can lead to incorrect conclusions. Additionally, if the data is missing or incomplete, you may not be able to get an accurate picture of the population that the dataset is supposed to represent. This can skew the results, making them less reliable or even completely useless. Therefore, data quality is an important consideration when evaluating statistics.

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Combining these, we have[tex]$6 \times 6 = \boxed{36}$[/tex] different arrangements of the numbers [tex]${}1,$ ${}2,$ ${}3,$ ${}4,$ ${}5,$ $6$[/tex] in a row where the sum of any adjacent numbers is even.

Products intended for exporting after processing into goods or processed products are referred to as "basic products," as are goods planned for export after processing. Samples 1 - 3 Samples 2 - 3.

We may start by noting that at minimum one of the neighboring numbers must be even for the sum of both numbers to be even. This means that a even numbers (2, 4, 6) as well as the odd numbers (1, 3, 5) should be arranged in the appropriate positions.

Let's start by thinking about the even positions. The second, fourth, and sixth places are the only even positions. We can choose any variant of the 3 even numbers to occupy these spots, giving us[tex]$3! = 6$[/tex] ways.

Let's now think about the unusual positions. The first, third, and fifth positions are the only ones that are odd. We have an additional [tex]$3! = 6$[/tex]ways to fill these spots by using any combination of the 3 odd numbers.

Consider the odd locations now. The first, third, and fifth places are the three odd positions. We have an additional[tex]$3! = 6$[/tex]ways by using any permutation of the three odd numbers to fill these positions.

Together, this give us [tex]$6 \times 6[/tex] = [tex]\boxed{36}$[/tex] different ways to arrange the numbers [tex]${}1,$ ${}2,$ ${}3,$ ${}4,$ ${}5,$ $6$[/tex]in a row so that the sum of any two adjacent numbers is even.

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Answer:the triangles are similar to each other by AA rule and EF = 8.

Step-by-step explanation:

the triangles are similar to each other by AA rule and EF = 8.

this should help if wrong sorry im in middle school so

Option C. 6,160 yards. To convert miles to yards, we multiply the number of miles by 1,760 (which is the number of yards in one mile). So, 3.5 miles x 1,760 yards/mile = 6,160 yards.

To convert miles to yards, we need to know that there are 1,760 yards in one mile. Therefore, to find out how many yards are in three and one-half miles, we need to multiply 1,760 by 3.5:

1,760 x 3.5 = 6,160

Therefore, there are 6,160 yards in three and one-half miles.

The other options provided are:

4,400 yards: This is equivalent to 2.5 miles, since 1 mile is 1,760 yards. Therefore, this is not the correct answer.

5,280 yards: This is equivalent to 3 miles, since 1 mile is 1,760 yards. Therefore, this is also not the correct answer.

7,040 yards: This is equivalent to 4 miles, since 1 mile is 1,760 yards. Therefore, this is not the correct answer.

So, the correct answer is 6,160 yards, which is the result of multiplying 1,760 yards by 3.5 miles.

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The example of 2×3 matrix A and a 3×2 matrix B such that, letting T(x)=Ax and U(x)=Bx is reflection Ab.

Matrices, the plural form of matrix, are the groupings of numbers, variables, symbols, or phrases in a rectangular table with varying rows and columns. These are rectangular arrays with specified operations such as addition, multiplication, and transposition. The elements of the matrix are the numbers or entries in it. The horizontal entries of matrices are referred to as rows, whereas the vertical elements are referred to as columns.

Let,

[tex]B = \left[\begin{array}{cc}0&1\\1&0&0&0\end{array}\right] , A = \left[\begin{array}{ccc}1&0&0\\0&1&0\\\end{array}\right][/tex]

Then,

[tex]AB =\left[\begin{array}{ccc}1&0&0\\0&1&0\\\end{array}\right] \left[\begin{array}{cc}0&1&1&0&0&0\\\end{array}\right] \\\\AB = \left[\begin{array}{cc}0&1&1&0\\\end{array}\right][/tex]

Therefore, Ab is reflection about y = x .

As U = Bx and T∘U    

A matrix is a rectangular array of integers, variables, symbols, or expressions that are defined for subtraction, addition, and multiplication operations. The number of rows and columns in a matrix determines its size (also known as the order of the matrix).          

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The phrase "He went to get milk but never came back" is often used as a humorous way to explain someone's absence or to imply that someone is unreliable or untrustworthy.

The phrase likely originates from a common experience where a child's parent, often their father, promises to go out to get something, like milk, but never returns. This can be a source of disappointment and confusion for the child, and the phrase has since been used in a joking manner to explain someone's failure to show up or fulfill a promise.

However, it is important to recognize that this experience can also be a source of trauma and should not be used to make light of someone's pain or loss.

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Answer:

$11.22

Step-by-step explanation:

20% of 56.09 = 1/5 of 56.09

Now, we divide 56.09 by 5 to get our answer which when rounded becomes 11.22.

Answer:

$11.22

Step-by-step explanation:

20% = 0.2

We Take

56.09 x 0.2 = $11.218

Round $11.218 will be $11.22

So, the tip amount is $11.22

The probability that the response indicated that the dog is small-sized given that they enjoyed the treat is 0.286 (or 2/7) in fraction in the lowest terms.

Bayes' theorem is used to update probabilities of a hypothesis or an event in light of new data or evidence. It is used to calculate the conditional probability of an event based on prior knowledge of the conditions that might be relevant to the event.In the given problem, we have to find the probability that the response indicated that the dog is small-sized given that they enjoyed the treat.

The probability that the dog is small-sized given that they enjoyed the treat is the conditional probability P(S|T), where S is the event that the dog is small-sized and T is the event that they enjoyed the treat. To find the value of P(S|T), we will use Bayes' theorem. Bayes' theorem states that P(S|T) = P(T|S) * P(S) / P(T) where P(T|S) is the probability that they enjoyed the treat given that the dog is small-sized, P(S) is the prior probability that the dog is small-sized, and P(T) is the probability that they enjoyed the treat.

P(S) = 3/7P(T|S) = 2/3P(T) = (2/3 * 3/7) + (1/4 * 4/7) = 18/84 + 4/28 = 1/3

(adding the probabilities of T given S and T given L)Therefore, P(S|T) = (2/3 * 3/7) / (1/3) = 2/7 = 0.285714...Rounding off to the nearest millionth, the probability is 0.286. Therefore, the probability that the response indicated that the dog is small-sized given that they enjoyed the treat is 0.286 (or 2/7) in fraction in the lowest terms.

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To find the derivative of the function f(x), we need to take the derivative of the natural logarithm (ln) of the function. We can do this by using the chain rule, which states that the derivative of the composition of two functions is equal to the derivative of the outer function times the derivative of the inner function.

The derivative of the outer function (ln) is 1/f(x), and the derivative of the inner function (14 - e-2x) is -2e-2x. So the derivative of f(x) is:

f'(x) = 1/f(x) × (-2e-2x)

f'(x) = 1/(ln(14 - e-2x)) × (-2e-2x)

f'(x) = -2e-2x/(14 - e-2x)

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we can use the notation P(A) and apply the definition of probability: P(A) = P(B), P (A ∩ B), and P (A ∪ B) if we have the necessary information.

I believe you meant to say, "Venn diagram". A Venn diagram is a type of graphical representation used to illustrate relationships between sets or groups of objects, concepts, or ideas. It consists of a series of overlapping circles or other closed shapes, with each circle representing a set or group and the overlapping areas representing the relationships or intersections between them.

by the question.

Let A be the event that a student is a cast member of Seussical, and let B be the event that a student is a cast member of Wizard of Oz. Then, we can define the following:

A ∩ B: The event that a student is a cast member of both Seussical and Wizard of Oz.

A ∪ B: The event that a student is a cast member of either Seussical, Wizard of Oz, or both.

A': The event that a student is not a cast member of Seussical.

B': The event that a student is not a cast member of Wizard of Oz.

Based on the information given, we do not know how many students are in each of these events, but we can still make some general observations. For example:

If A and B have no students in common (i.e., A ∩ B = ∅), then the number of students in A ∪ B is equal to the sum of the number of students in A and the number of students in B.

If some students are in both A and B (i.e., A ∩ B is not empty), then the number of students in A ∪ B is equal to the sum of the number of students in A, the number of students in B, and the number of students in A ∩ B. In other words, some students are counted twice when we add up the number of students in A and the number of students in B, so we need to subtract the number of students in A ∩ B to avoid double-counting.

We do not know whether the events A and B are mutually exclusive (i.e., whether A ∩ B = ∅) or not. If they are mutually exclusive, then P(A ∩ B) = 0, and we can use the addition rule of probability to find P (A ∪ B) = P(A) + P(B). If they are not mutually exclusive, then we need to use the general addition rule of probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Finally, if we want to find the probability that a student chosen at random is a cast member of Seussical,

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Answer: The cost of painting is Rs.254.4

Step-by-step explanation:

let  l and b be the sides

        b=3m 20 cm=3.20m    

                                          = 5.30×3.20

                                           =16.96m²

We are given that, the measures of sides of blackboard are 5 m 30 cm and 3m 20 cm.

__________________________________________

[tex] \bf \implies5 m + 30 cm \\ [/tex]

[tex] \sf \implies 5 m + \dfrac{30}{100}m \\ [/tex]

[tex] \sf \implies 5 m + \dfrac{3\cancel{0}}{10\cancel{0}}m \\ [/tex]

[tex] \sf \implies 5 m + 0.3 m \\ [/tex]

[tex]\purple{ \bf \implies 5.3~ m } \\ [/tex]

[tex] \bf \implies3 m + 20 cm \\ [/tex]

[tex] \sf \implies 3 m + \dfrac{20}{100}m \\ [/tex]

[tex] \sf \implies 3 m + \dfrac{2\cancel{0}}{10\cancel{0}}m \\ [/tex]

[tex] \sf \implies 3 m + 0.2 m \\ [/tex]

[tex] \purple{\bf \implies 3. 2 ~m} \\ [/tex]

_______________________________________________

[tex] \pink{\frak{\implies Area _{(Blackboard) }= Length \times Breadth ~m^2}} \\ [/tex]

[tex] \sf \implies Area _{(Blackboard) } = 5.3 \times 3.2 ~m^2 \\ [/tex]

[tex] \sf \implies Area _{(Blackboard) } = 16.96 m^2 \\ [/tex]

[tex] \sf \implies 15 \times 16.96 \\ [/tex]

[tex] \pink{\sf \implies Rs ~254.4 } \\ [/tex]

The length of the third side must fall between 8 and 13. This is because the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.

Answer:

1. n=6

2. n=19

3. n=15

4. n=20

5. n=40

6. n=16

7. n=19

8. n=6

9. n=2.4

10. n=30

Step-by-step explanation:

remember the priorities :

1. brackets

2. exponents

3. multiplications and divisions

4. additions and subtractions

inside every category you go usually from left to right, but you can use the commutative property where applicable.

4 + 5 - 3 = 4 + 5 - 3 = 4 + 5 - 3 = 4 - 3 + 5 = 6

8×3 - 5 = 24 - 5 = 19

15/3 + 10 = 5 + 10 = 15

8 + 3 - 1×2 = 8 + 3 - 2 = 9

14×3 - 2 = 42 - 2 = 40

(3 + 5)×2 = 8×2 = 16

10 + 7 + 2×1 = 10 + 7 + 2 = 19

16/8 + 4 = 2 + 4 = 6

7 + 5/5 = 7 + 1 = 8

20/4×6 = 5×6 = 30

Answer:

not sure if this sign � was important did it the best way I could

Step-by-step explanation:

To solve this problem graphically, we will first set up a system of inequalities based on the given information:

x ≥ 8 (Fatoumata must work at least 8 hours lifeguarding)

y ≤ 12 - x (Fatoumata can work at most 12 total hours)

15x + 10y ≥ 140 (Fatoumata must earn a minimum of $140)

To graph these inequalities, we can plot the points (8,0), (12,0), and (0,14) on a coordinate plane and draw lines connecting them. The line between (8,0) and (12,0) represents the constraint on the number of hours Fatoumata can work, while the line between (8,0) and (0,14) represents the constraint on the amount of money she must earn. The shaded region that satisfies all three inequalities is the feasible region.

To find one possible solution, we can pick any point within the feasible region. One such point is (8,6), which represents working 8 hours lifeguarding and 6 hours tutoring. This point satisfies all three inequalities:

x ≥ 8 is true since x = 8

y ≤ 12 - x is true since y = 6 ≤ 12 - 8

15x + 10y ≥ 140 is true since 15(8) + 10(6) = 180 ≥ 140

Therefore, one possible solution is for Fatoumata to work 8 hours lifeguarding and 6 hours tutoring to earn at least $140 while not exceeding 12 total hours worked.

Over a year, the Blue Valley Glacier typically moves about the same distance as the Silver Lake Glacier, and Blue Valley has the same variability in its annual movement compared to Silver Lake.

The median annual movement of the Blue Valley Glacier is 300.2 feet, and the interquartile range is 14 feet.

The interquartile range indicates the spread of the data within the middle 50% of the data

So we know that the annual movement of the Blue Valley Glacier falls within a range of 300.2 ± 7 feet (i.e. 293.2 to 307.2 feet)

Similarly, the median annual movement of the Silver Lake Glacier is 300.4 feet, and the interquartile range is also 14 feet

So the annual movement of the Silver Lake Glacier also falls within a range of 300.4 ± 7 feet (i.e. 293.4 to 307.4 feet)

Since the ranges for both glaciers overlap and have the same size, we can conclude that they typically move about the same distance over a year, and that the variability in the annual movement of Blue Valley is comparable to that of Silver Lake.

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The formula relating I and P is I = kP

a) When P= $800, then I = $48

b) When I = $72, then P = $1200

If the interest $I on a loan of $P for a year at a rate of 6% varies directly as the loan, we can write:

I = kP

where k is a constant of proportionality. To find the value of k, we can use the given information that the interest rate is 6%, or 0.06 as a decimal. We know that when P = 100, the interest I = 0.06 × 100 = 6. Therefore:

I/P = 6/100 = 0.06 = k

Now we can use this value of k to answer the given questions,

a) When P = 800, the formula relating I and P is:

I = kP

I = 0.06 × 800

I = 48

Therefore, the interest on a loan of $800 for a year at a rate of 6% is $48.

b) When I = 72, the formula relating I and P is:

I = kP

72 = 0.06P

Solving for P:

P = 72/0.06

P = 1200

Therefore, a loan of $1200 for a year at a rate of 6% would have an interest of $72.

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