A ladder leans against the side of a house. The angle of elevation of the ladder is 61 degrees , and the top of the ladder is 5 m above the ground. Find the distance from the bottom of the ladder to the side of the house. Round your answer to the nearest tenth.

We need to show that if we choose right triangle with the interior angle θ, the ratios of the sides will be the same as shown in below figure.

The mathematical subject of trigonometry is the study of the connections between the angles and sides of triangles. It entails investigating trigonometric functions like sine, cosine, and tangent, which examine the relationship between a triangle's sides' lengths and its angles.

Let's consider a right triangle with acute angles α and β, as shown below:

To relate the sides of this triangle, we can use the Pythagorean theorem:

a² + b² = c²

Now, let's define the six trigonometric functions in terms of the sides of the triangle:

Sin(θ) = a/c

Cos(θ) = b/c

tan(θ) = a/b

csc(θ) = c/a

sec(θ) = c/b

cot(θ) = b/a

We want to show that these functions are well-defined, i.e., they do not depend on the particular triangle we choose. To do this, we need to show that if we choose another right triangle with the same interior angle θ, the ratios of the sides will be the same.

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Answer:

please insert question

Step-by-step explanation:

Answer:

4 students

Step-by-step explanation:

Let's call the total number of students in the school "x". We know that a certain percentage of them had over 80% in math, and the rest (100% - that percentage) had 80% or less.

Let's call the percentage of students who had over 80% "p". Then, we can set up the following equation:

p% of x + (100% - p%) of x = x

We can simplify this to:

p/100 * x + (100 - p)/100 * x = x

Multiplying both sides by 100 to get rid of the denominators, we get:

px + (100 - p)x = 100x

Simplifying further:

px + 100x - px = 100x

100x = 465

x = 465/100 = 4.65 (rounded to two decimal places)

So the total number of students in the school is approximately 4.65. However, we can't have a fraction of a student, so let's round up to the nearest whole number and assume there are 5 students in the school.

Now we can use the information given to find the number of students who had over 80%:

"of the students had over 80% in math"

implies that

p% of x = number of students who had over 80%

So if we plug in the values we have:

p% of 5 = number of students who had over 80%

Simplifying:

0.01p * 5 = number of students who had over 80%

0.05p = number of students who had over 80%

We don't know the value of p, but we can solve for the number of students who had over 80% for different values of p. For example:

If p = 90, then:

0.05(90) = 4.5

So 4.5 students had over 80%. Since we can't have half a student, we can assume that 4 students had over 80%.

Alternatively, we can solve for p using the information given:

"of the students had over 80% in math"

implies that

p% of x = number of students who had over 80%

So:

p% of x = number of students who had over 80%

p% of 5 = number of students who had over 80%

0.01p * 5 = number of students who had over 80%

0.05p = number of students who had over 80%

We know that the number of students who had over 80% is some integer value between 0 and 5, inclusive. We can test different values of p within this range to see if they give us an integer solution:

If p = 90, then:

0.05(90) = 4.5

This is not an integer solution, so p = 90 is not the correct answer.

If p = 80, then:

0.05(80) = 4

This is an integer solution, so p = 80 is the correct answer. Therefore, 4 students had over 80%.

The triangles which are similar or congruent discussed below.

If the triangles are similar then the corresponding side ratios are equal.

1. 6/9 = 2/3

9/13.5= 2/3

8/12= 2/3

Thus, the triangle are similar.

2. The second case is neither similar or congruent.

3. The pair has one right angle common and one side measure 6 unit.

So, the triangle are neither similar or congruent.

4. The triangles have two angles equal.

so, the triangle are similar.

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Step-by-step explanation:

The answer is:

- Control of significant inventory balances

This is because just-in-time processing is a system that emphasizes on producing goods or services at the exact time they are needed, without accumulating inventory. Therefore, it does not prioritize the control of significant inventory balances. The other options are benefits of just-in-time processing.

As per the given equation the correct time, t, that it took for the ball to hit the ground is t = 3 seconds.

An equation is a mathematical statement that expresses the equality between two expressions. Equations typically include variables, which are symbols that represent unknown or varying quantities, and constants, which are known values.

An equation can be written in various forms, depending on the type of equation and the context in which it is used. Some common forms of equations include linear equations, quadratic equations, and polynomial equations.

For example, the equation x + 5 = 10 is a linear equation that has one variable, x. It can be solved by subtracting 5 from both sides of the equation to get x = 5.

A quadratic equation, such as x² + 2x + 1 = 0, has a variable raised to the second power. It can be solved using the quadratic formula or by factoring.

To solve the equation 0=−16t²+32t+48, Greg factored out a common factor of -16 from all three terms to get:

0 = -16(t² - 2t - 3)

Then, he factored the quadratic expression inside the parentheses as:

0 = -16(t - 3)(t + 1)

This gives us two solutions for t: t = 3 and t = -1.

However, we can discard the solution t = -1, since time cannot be negative in this context. Therefore, the correct time, t, that it took for the ball to hit the ground is t = 3 seconds.

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y-intercept - (0, 4)

slope - +2

Domain - input values

Range - output values

Is this graph increasing, decreasing, or both? - Increasing

x-intercept - (-2, 0)

Answer: Mean = 1.15       Median = 1

Step-by-step explanation: add all numbers up and then divide it by 4 which will give you the mean.

Median: 0.9 and 1.1 are the 2 middle numbers in which you will add together then divide by 2 to get the exact middle value which is 1 exactly.

the final result of the double integral is `(4/3)*a. we have to Integrate the inner integral with respect to z.

what is inner integral ?

An inner integral is a mathematical term that refers to the integral function that is evaluated first in a double integral.

In the given question,

To solve this double integral, we will use the following steps:

Integrate the inner integral with respect to z.

Evaluate the result of the inner integral at  upper and lower limits of z.

Substitute the result of the inner integral into the outer integral and integrate with respect to y.

Evaluate the result of the outer integral at the upper and lower limits of y.

Simplify the expression.

Now, let's apply these steps to solve the given double integral:

Integrate the inner integral with respect to z:

∫(x²*z + y²*z + z³) dz = x²/2*z² + y²/2*z² + z^4/4 + C

where C is the constant of integration.

Evaluate the result of the inner integral at the upper and lower limits of z:

(x²/2*(a²-x²-y²)¹⁵ + y²/2*(a²-x²-y²)¹⁵ + (a²-x²-y²)²/4)

- (x²/2*(-a²+x²+y²)¹⁵ + y²/2*(-a²+x²+y²)¹⁵ + (-a²+x²+y²)²/4)

Substitute the result of the inner integral into the outer integral and integrate with respect to y:

markdown

∫[(x²/2*(a²-x²-y²)¹⁵ + y²/2*(a²-x²-y²)¹⁵ + (a²-x²-y²)²/4)

- (x²/2*(-a²+x²+y²)¹⁵ + y²/2*(-a²+x²+y²)¹⁵ + (-a²+x²+y²)²/4)] dy

Evaluate the result of the outer integral at the upper and lower limits of y:

= ∫[(x²/2*(a²-x²-y²)¹⁵ + y²/2*(a²-x²-y²)¹⁵ + (a²-x²-y²)²/4)- (x²/2*(-a²+x²+y²)¹⁵ + y²/2*(-a²+x²+y²)¹⁵ + (-a²+x²+y²)²/4)] dy

from y = -sqrt(a²-x²) to y = sqrt(a²-x²)

= (2/3)*x²*(a²-x²)¹⁵ + (2/3)*(a²-x²)²⁵

- (2/3)*x²*(-a²+x²)¹⁵ + (2/3)*(-a²+x²)²⁵

Simplify the expression:

= (4/3)*a³ - (4/3)*a*x²

Therefore, the final result of the double integral is `(4/3)*a

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The total number of sales after 8 weeks through which the given condition is satisfied is 4060 sales.

Explain the process through which sales is identified?

There are several mathematical formulas that can be used in sales to calculate different metrics such as profit, revenue, and discounts. Here are some examples:

Profit formula: Profit = Revenue - Cost

This formula is used to calculate the profit earned from the sale of a product or service. Revenue is the total amount of money generated from sales, and cost includes all the expenses associated with producing and selling the product.

Gross margin formula: Gross Margin = (Revenue - Cost of Goods Sold) / Revenue

This formula is used to calculate the gross margin percentage, which is the difference between revenue and the cost of goods sold, divided by revenue. This metric represents the percentage of revenue that is left over after accounting for the cost of producing the product.

Discount formula: Discount = List Price x Discount Rate

This formula is used to calculate the amount of discount applied to the list price of a product. The discount rate is expressed as a percentage, and is multiplied by the list price to determine the amount of the discount.

Markup formula: Markup = (Selling Price - Cost) / Cost

This formula is used to calculate the markup percentage, which is the difference between the selling price and the cost of producing the product, divided by the cost. This metric represents the percentage increase in price that is added to the cost to arrive at the selling price.

According to the given information:

Given, 350 sales in the first week and increases the number of sales by 45 each week.

Based on the given condition fromulate:

⇒ 8 x ( 2 x 350 + 45 x (8-1)) / 2

After solving the above relation we get,

⇒ 4 x (2x350 + 45 x 7)

⇒ 4 x (700 + 315)

⇒ 4 x 1015

⇒ 4060.

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The limit of the given sum as n approaches infinity is 0.

We have the limit as n approaches infinity of the sum from i = 1 to n of 1/(n+i). We can rewrite this sum using the hint provided as:

[tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex](1/ n + i) = [tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex] (1/n) * (1 + i/n)

To find the limit, we need to take the limit of the Riemann sum as n approaches infinity. This is equivalent to taking the limit of the area of n rectangles under the curve y=1/x as n approaches infinity.

As n becomes very large, the width of each rectangle becomes very small, and the height of each rectangle approaches 1/n. Therefore, the area of each rectangle approaches zero.

We can then express the limit as an integral:

[tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex](1/ n + i) =[tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex] (1/n) * (1 + i/n) =[tex]\lim_{n \to \infty} \int ^1 _{1+1/n}[/tex] 1/x dx

Evaluating this integral gives:

[tex]\lim_{n \to \infty} \int ^1 _{1+1/n}[/tex] 1/x dx = [tex]\lim_{n \to \infty}[/tex] ln(1+1/n) = ln(1+0) = 0

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The Distributive Property of multiplication indicates that the area of the rectangular deck is; Area = 5.5·x + 46.75 square meters

The Distributive Property of multiplication states that a × (b + c) = (a·b + a·c).

The increase in the length of the deck = x meters

The area of the deck 5.5·(x + 8.5)

The width of the deck = 5.5 meters

The length of the deck = (x + 8.5) meters

The expression that represents the area of the whole deck obtained using the Distributive Property is therefore;

Area = 5.5·(x + 8.5) = 5.5 × (x + 8.5)

Distributive Property indicates that we multiply each term in the expression within the bracket symbol by 5.5 as follows;

Area = 5.5 × x + 5.5 × 8.5 = 5.5·x + 46.75

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If the recipe requires 3(1/2) pounds of flour for every (3/4) cups of sugar, then for 1 cup of sugar , Joseph should use 4.67 pounds of flour.

A "Proportion" is defined as a statement that two fractions are equal. It expresses the relationship between two quantities that are in the same ratio.

To solve the problem, we set up a proportion to relate the amount of flour to the amount of sugar:

We know that, recipe requires 3(1/2) pounds of flour for every (3/4) cups of sugar,

Which means,

⇒ (3.5 pounds of flour)/(0.75 cups of sugar) = (x pounds of flour)/(1 cup of sugar),

where x is = amount of flour needed for 1 cup of sugar.

We "cross-multiply" and simplify;

⇒ 3.5 × 1 = 0.75 × "x" pounds of flour,

⇒ 3.5/0.75 = x,

⇒ x ≈ 4.67 pounds of flour,

Therefore, Joseph should use 4.67 pounds of flour for 1 cup of sugar.

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I can assist you in determining the r-coordinate of vertex K* following the reflection if you provide me the coordinates of vertex K in the original quadrilateral.

It is possible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical expression (such as addition, subtraction, multiplication or division, etc.) include numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression, often known as an algebraic expression, is any mathematical statement that contains variables, numbers, and an arithmetic operation between them. For instance, the word m in the given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as does the variable m in the expression 4m + 5.

I can assist you in determining the r-coordinate of vertex K* following the reflection if you provide me the coordinates of vertex K in the original quadrilateral.

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The length of side c is approximately 4.62 units.

Calculating the value of side c can be done by applying the Law of Sines on two angles and two sides.

Initially, determining angle C is necessary:

The corresponding formula for Angle C: 180° - (Angle A + Angle B) evaluates to 90° after substituting in reference values; 60°, and 30° respectively.

Implementing the Law of Sines culminates in this expression:

a/sin(A) = b/sin(B) = c/sin(C)

Replace with known angles and evaluate as follows:

Since the measures are known;

c/sin(C) = a/sin(A).

Subsequently,

by simple algebra c= a * sin(C) / sin(A) leads to desired outcome.

Putting relevant points into the equation above gives:

c = 4 * sin(90°) / sin(60°)

By using sine values from a calculator:

the preceding expression becomes,

c = 4 * 1 / (sqrt(3)/2)

After solving for c:

c = 8 / sqrt(3)

Using square root's rationalization: multiply numerator, and denominator by √3 resulting in

c = (8 * sqrt(3)) / 3

The length of side c is approximately 4.62 units.

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Let's assume we have a triangle with angles A and B and sides a and b, where angle A is opposite to side a, and angle B is opposite to side b. Given angle A = 60°, angle B = 30°, side a = 4 units, and side b = 2 units, find the length of side c.

The graph is positive and increases as the value of x increases.

The behavior of the graph of y = √(4x) is determined by substituting some value of x into the function and check the corresponding value of y.

When x = 0, the value  of y is calculated as;

y = √(4(0))

y = 0

When x = 1, the value of y is calculated as;

y = √(4(1))

y = 2

When x = 4, the value of y is calculated as;

y = √(4(4))

y = 4

When x = 9,  the value of y is  calculated as;

y = √(4(9))

y = 6

From the data above, the value of y increases as x increases, although not at equal increment.

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Answer:

780

Step-by-step explanation:

In this example we will call Original Price = y

72%=0.72

218.40=0.72*y

1-0.72=0.28

218.4÷0.28=780

Answer:

Point A (-3,3)

Point B about (3,-2.75)

Point C (-4,-4)

Point D (1,0)

Step-by-step explanation:

Answer:

reverse or flip

Answer:

The new prism has 9 times the surface area of the original prism

Step-by-step explanation:

There are three ways in which you can answer this question.

The hard way:

The surface area of a rectangular prism is given by
A = 2(LW+ LH+ WH)
where L= length, W= width and H = height)

If we were to triple the sides we would get the new side measures as
3L, 3W, 3H

New surface area becomes:
A' = 2 (3L · 3W + 3L · 3H · 3W· 3H)

A' = 2(9LW + 9 LH + 9 WH)

Factoring out 9 from the brackets we get

A' = 2 · 9 (LW+ LH+ WH)

A'/A = 2 · 9 (LW+ LH+ WH) /2(LW+ LH+ WH)

The common term 2(LW+ LH+ WH) cancels out from numerator and denominator leaving 9 as the answer

A smarter and easy way of doing this

A cube is nothing but a rectangular prism with all sides equal. Let a be the length of a side of the cube

A cube has 6 sides. The surface area of each side = a x a = a²

So total surface area A = 6a²

If each side is tripled, each side becomes 3a.
New surface area A' = 6 (3a)² = 6 (9a²)

A'/A = 6 (9a²)/6(a²) = 9

An even easier way

Again we take a cube. But instead of using a variable, let's assign the side of the cube a length of 1 unit

Surface area A = 6 · 1² = 6

After tripling each side becomes 3 units long
New surface area A' = 6 · 3² = 6.9 = 54

A'/A = 54/6 = 9

Choose whichever method you feel comfortable with

Answer:

3 1/8 or 25/8

Step-by-step explanation

This is because you are trying to find how much longer milan took. So we subtract 3 1/4 - 1/8

To do this convert 3 1/4 to 3 2/8 -1/8

The two points on the line are (5, -9/5) and (-5/3, 3).

The other point on the line is (0, -12/7).

The equation [tex]\frac{m}{5} +\frac{n}{3} =0[/tex] can be rewritten as:

3m + 5n = 0

When m=5:

3m + 5n = 0

3(5) + 5n = 0

n = -9/5

Simplify the above equation,

When n=3:

3m + 5n = 0

3m + 5(3) = 0

m = -5/3

the two points on the line are (5, -9/5) and (-5/3, 3)

The equation [tex]\frac{m}{10} -\frac{7n}{6} =4[/tex] can be simplified as follows:

m/10 - 7n/6 = 4

m/10 = 7n/6 + 4

m = 70n/6 + 40

m = 35n/3 + 20

When n=0:

m = 35n/3 + 20

m = 20

So, the point on the line is (20, 0).

When m=0:

35n/3 + 20 = 0

35n/3 = -20

n = -12/7

So, the other point on the line is (0, -12/7).

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Answer:

1.fraction=1/4

percentage is 0.25

likelihood=unlikely

2.fraction=5/16

percentage=0.3125

likelihood=unlikely

3.fraction=11/16

percentage=0.6875

likelihood=likely

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Answer: Yes, the triangles are congruent.

Step-by-step explanation:

According to the Side-Angle-Side (SAS) congruence theorem, if two triangles have two pairs of corresponding sides that are congruent and the included angle between those sides is also congruent, then the triangles are congruent.

In this case, we know that four parts (angles and sides) of one triangle are congruent to the corresponding four parts of another triangle. This means that two pairs of corresponding sides are congruent (since corresponding sides are equal) and the included angles between those sides are also congruent (since corresponding angles are equal). Therefore, the triangles satisfy the conditions of the SAS congruence theorem and are congruent.

It's important to note that this applies only to two triangles with exactly the same four congruent parts. If there is even one part that is not congruent between the two triangles, they cannot be proven to be congruent just by these means alone.

Answer:

If you know that 4 parts (angles and sides) of one triangle are congruent to the corresponding 4 parts of another triangle, then the triangles are congruent by the Side-Angle-Side (SAS) Congruence Postulate. This postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

The SAS postulate is one of five ways to prove that two triangles are congruent. The other four ways are Angle-Side-Angle (ASA), Side-Side-Side (SSS), Hypotenuse-Leg (HL), and Reflexive Property of Congruence.

The z score of 0.166 square units of the standard normal distribution is to leave z is -0.999.

It is clear that 0.166 square units of the standard normal distribution is to the left of z. This means that the area under the standard normal distribution curve to the left of z is 0.166.

Using a standard normal distribution table or calculator, we can find the z-score corresponding to this area. For example, using a calculator, we can use the inverse normal cumulative distribution function (also called the probit function) to find the z-score:

invNorm(0.166) ≈ -0.999

Therefore, the z-score corresponding to 0.166 square units of the standard normal distribution is to the left of z is approximately -0.999.

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On differentiating, the derivative of the functions at x = a, is

(i) f'(a) = 14/(a+9)²,

(ii) f'(a) = [tex]-\frac{1}{2(a+1)^{\frac{3}{2} } }[/tex].

Part (i) : To find the derivative of f(t) = (2t+4)/(t+9) at x = a, we can use the quotient rule:

The "quotient-rule" is defined as a formula which is used in finding  derivative of a function which is quotient of two other functions.

The formula is : (f/g)' = (f'g - g'f)/g²,

where f' , g' = derivatives of functions "f" and "g", respectively.

So, f'(t) = [(t+9)×(2) - (2t+4)×(1)] / (t+9)²,

Substituting t = a,

We get,

⇒ f'(a) = [(a+9)(2) - (2a+4)(1)] / (a+9)²,

⇒ f'(a) = (2a+18 - 2a-4) / (a+9)²,

⇒ f'(a) = 14/(a+9)²,

So, the derivative of f(t) at x = a is f'(a) = 14/(a+9)².

Part (ii) : To find the derivative of the function, f(x) = 1/√(x+1) at x = a, we use the chain rule:

The "Chain-Rule" is defined as a formula which is used in finding derivative of a "composite-function".

So, f'(x) = [tex]-\frac{1}{2(x+1)^{\frac{3}{2} } }[/tex] × (1)

Substituting x = a,

We get,

⇒ f'(a) =  [tex]-\frac{1}{2(a+1)^{\frac{3}{2} } }[/tex] ,

Therefore, the derivative of f(x) at x = a is f'(a) = [tex]-\frac{1}{2(a+1)^{\frac{3}{2} } }[/tex].

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The given question is incomplete, the complete question is

Find the derivative of the given functions at "x=a".

(i) f(t) = (2t+4)/(t+9),

(ii) f(x) = [tex]\frac{1}{\sqrt{x+1} }[/tex]  1/√(x+1)

To make m the subject of the formula in the equation E = mgh + 1 / 4mv² is m = E / mgh + 1 / 4mv²

The equation E = mgh + 1 / 4mv².

An equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

A variable is a number represented with a letter in an equation. m is the variable to make the subject of the formula.

Therefore,

E = mgh + 1 / 4mv²

factorise the equation on the right side

E = m(gh + 1 / 4v²)

Divide both sides of the equation by mgh + 1 / 4mv²

Hence,

m = E / mgh + 1 / 4mv²

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Answer:

  (x, y, z) = (-20, 10, -19)

Step-by-step explanation:

You want the values of x, y, and z given a rhombus with sides marked (-2x+6), (5y-4), (-2z+8), and 46.

The side lengths of a rhombus are all the same. This tells us ...

Each of these 2-step equations can be solved by subtracting the unwanted constant and dividing by the coefficient of the variable.

  x = (46 -6)/(-2) = -20

  y = (46 -(-4))/5 = 10

  z = (46 -8)/-2 = -19

The values of x, y, and z are -20, 10, and -19, respectively.

The required balance after 10 years with continuous compounding at a 7% interest rate would be approximately $1007.

To calculate the balance after 10 years with a continuous compounding interest rate of 7%, we can use the formula for continuous compound interest:

[tex]F = P * e^{rt}[/tex]

Where:

F is the future balance or the final amount

P is the principal amount (initial deposit)

e is the base of the natural logarithm (approximately 2.71828)

r is the interest rate (expressed as a decimal)

t is the time in years

In this case, the initial deposit (principal) is $500, the interest rate is 7% (0.07 as a decimal), and the time is 10 years. Plugging these values into the formula, we get:

[tex]F = 500 * e^{0.07 * 10}[/tex]

Using a calculator, we can evaluate e^(0.07 * 10) ≈ 1.96728. Multiplying this by 500 gives us:

[tex]F=500 * 1.96728[/tex]
F = $1007

Therefore, the balance after 10 years with continuous compounding at a 7% interest rate would be approximately $1007.

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