One leg of a right triangle is 7 and the hypotenuse is 10. What is the length of the
other leg?

To find the length of the segment between the given points, we can use the distance formula. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
\[d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\]

Let's apply this formula to the given points: $(2a, a-4)$ and $(4, -1)$.

The distance between these two points is $2\sqrt{10}$ units. So we have:
\[2\sqrt{10} = \sqrt{{(4 - 2a)^2 + (-1 - (a-4))^2}}\]

Simplifying the equation, we get:
\[4\sqrt{10} = \sqrt{{(4 - 2a)^2 + (-5 - a)^2}}\]

Squaring both sides of the equation, we have:
\[160 = (4 - 2a)^2 + (-5 - a)^2\]

Expanding the equation, we get:
\[160 = 16 - 16a + 4a^2 + 25 + 10a + a^2\]

Combining like terms, we have:
\[0 = 5a^2 - 6a + 1\]

Now, we can solve this quadratic equation for the possible values of $a$.

Factoring the equation, we have:
\[0 = (5a - 1)(a - 1)\]

Setting each factor equal to zero and solving for $a$, we get:

\[5a - 1 = 0 \quad \Rightarrow \quad a = \frac{1}{5}\]
\[a - 1 = 0 \quad \Rightarrow \quad a = 1\]

Therefore, the possible values for $a$ are $\frac{1}{5}$ and $1$. The product of these values is:
\[\left(\frac{1}{5}\right) \cdot 1 = \frac{1}{5}\]

So, the product of all possible values for $a$ is $\frac{1}{5}$.

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Answer:487.5

Step-by-step explanation:

if it is a 35% discount it means you are trying to find 65% of the number because 65+35=100%

to find 65%, you find 5% first, then multiply by 13 to get 65%.

to find 5%, you divide by 20.

750/20=37.5.

37.5*13=487.5

Answer:

the reason is they are equal to each other because they are both right angle triangle

Lucy has $7 less than Kristine and $5 more than Nina together, the three have $35. Lucy has $11.

Let's denote the amount of money that Kristine has as K, the amount of money that Lucy has as L, and the amount of money that Nina has as N.

According to the given information, we can form two equations:

Lucy has $7 less than Kristine: L = K - 7

Lucy has $5 more than Nina: L = N + 5

We also know that the three of them have a total of $35: K + L + N = 35

We can solve this system of equations to find the values of K, L, and N.

Substituting equation 1 into equation 3, we get:

K + (K - 7) + N = 35

2K - 7 + N = 35

Substituting equation 2 into the above equation, we get:

2K - 7 + (L - 5) = 35

2K + L - 12 = 35

Since Lucy has $7 less than Kristine (equation 1), we can substitute K - 7 for L in the above equation:

2K + (K - 7) - 12 = 35

3K - 19 = 35

Adding 19 to both sides:

3K = 54

Dividing both sides by 3:

K = 18

Now we can substitute the value of K into equation 1 to find L:

L = K - 7

L = 18 - 7

L = 11

Finally, we can find the value of N by substituting the values of K and L into equation 3:

K + L + N = 35

18 + 11 + N = 35

N = 35 - 18 - 11

N = 6

Therefore, Lucy has $11.

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The value of the missing angles b and c are 71° and 71° respectively

An equation is an expression that shows the relationship between two or more numbers and variables using mathematical operations. An equation can be linear, quadratic, cubic and so on, depending on the degree of the variable.

From the diagram:

b + 109 = 180° (angle in a straight line)

b = 71°

b = c (opposite angles are equal)

b = c = 71°

The value of b and c are 71° and 71° respectively

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Answer:

Step-by-step explanation:

Let O be the center of the circle, and let x be the length of the radius of the circle. Since segment DE is a minor arc, it subtends an angle of less than 180 degrees at the center of the circle, and its length is given by:

length of DE = (angle AOB / 360) * 2 * pi * x

We are given that the length of DE is 52 cm, and we know that angle AOB is 2 times angle ACB, since these angles subtend the same arc. Therefore, we have:

angle AOB = 2 * angle ACB

We can use the law of cosines to find angle ACB:

cos(ACB) = (AB^2 + BC^2 - AC^2)/(2 * AB * BC)

cos(ACB) = (25 + 64 - 81)/(2 * 5 * 8)

cos(ACB) = -1/8

Since angle ACB is acute, we have:

ACB = arccos(-1/8)

ACB = 100.14 degrees (rounded to two decimal places)

Therefore, angle AOB is twice this angle, or:

angle AOB = 200.28 degrees

Substituting the values of (angle AOB, x) into the equation for the length of DE, we get:

52 = (200.28 / 360) * 2 * pi * x

Simplifying and solving for x, we get:

x = 26 / pi

The circumference of the circle is given by:

circumference = 2 * pi * x

Substituting the value of x, we get:

circumference = 2 * pi * (26 / pi) = 52 cm

Therefore, the circumference of circle F is 52 cm.

The integral expression of the volume of the solid generated when r is roated about the horizantal line y=6 is, V = π∫[f(y)]^2 dy

Let's consider the region bounded by the function r=f(y) and the horizontal line y=6 in the xy-plane, where r represents the distance between the y-axis and a point on a curve.

When this region is rotated around the horizontal line y=6, it generates a solid whose volume can be expressed as an integral in terms of y as follows:

V = π∫[f(y)]^2 dy

Here, the limits of integration are determined by the range of y-values for which the curve exists within the region bounded by the x-axis and the line y=6.

Note that we have not evaluated this integral yet since we do not have enough information about the specific function f(y).

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The number of senior, adults and children tickets sold are 22, 76 and 66 respectively.

The museum sells three types of tickets, including those for seniors, adults and children.

On Tuesday morning the  museum sold 164 tickets. Therefore,

Let

x = number of senior ticket sold.

The number of adult tickets sold was three times as many as senior tickets sold.

The number of child tickets sold was 10 more than the number of adult tickets sold. Therefore,

Total ticket sold = 3x + 10 + 3x + x

164 = 7x + 10

154 = 7x

x = 154 / 7

x = 22

Therefore,

number of senior ticket sold = 22

number of adult tickets sold = 3 (22) = 66

number of child tickets sold = 10 + 3(22) = 76

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To show that `(n + 3)7 ∈ Θ(n7)`, we need to prove that `(n + 3)7 = Θ(n7)`.This can be done by showing that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)` .Now, let's prove the two parts separately:

Proof for `(n + 3)7 = O(n7)`.

We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≤ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≤ n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + n7
≤ 2n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6
≤ 2n7 + 84n6 + 441n5 + 2205n4 + 10395n3 + 45045n2 + 153609n + 729
```Thus, we can take `c = 153610` and `k = 1` to satisfy the definition of big-Oh notation. Hence, `(n + 3)7 = O(n7)`.Proof for `(n + 3)7 = Ω(n7)`We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≥ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≥ n7
```Thus, we can take `c = 1` and `k = 1` to satisfy the definition of big-Omega notation. Hence, `(n + 3)7 = Ω(n7)`.

As we have proved that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)`, therefore `(n + 3)7 = Θ(n7)`.Thus, we have shown that `(n + 3)7 ∈ Θ(n7)`.From the proof, we can see that we used the Binomial theorem to expand `(n + 3)7` and used algebraic manipulation to bound it from above and below with suitable constants. This technique can be used to prove the time complexity of various algorithms, where we have to find the tightest possible upper and lower bounds on the number of operations performed by the algorithm.

Hence, we have shown that `(n + 3)7 ∈ Θ(n7)` for non-negative integer n.

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Answer: 3054 in^3

Step-by-step explanation: The formula for the volume of a sphere is V = 4/3 pi r^3, where r is radius.

Since the diameter is 18in, the radius would be 1/2 or 9in, and plugged in V is

V = 4/3(3.1415)(9)^3

Answer:

1/6*π*18^3≈3053.62806

Step-by-step explanation:

Answer:the first one is d and the third one is 65% and I think the second one is 5 hours

Step-by-step explanation:

(PLZ READ )all you have to do to get the answer for the first one is divide for example on D it says 1 and then 5 and 2 and 10 first you divide 1 by 5 then 2 by ten and if they ALL have the same answer then they are proportional even if one does not have the same as All the rest it is not proportional.

and for the third one you can search it up

Using the trapezoidal rule with n = 5, the approximation for the integral of 5cos(x) from 0 to π is approximately 7.42. Using Simpson's rule with n = 4, the approximation for the integral of cos(x) from 0 to π/2 is approximately 1.02.

The trapezoidal rule is a numerical method used to approximate definite integrals. With n = 5, the interval [0, π] is divided into 5 subintervals of equal width. The formula for the trapezoidal rule is given by h/2 * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)], where h is the width of each subinterval and f(xi) represents the function evaluated at the points within the subintervals.Applying the trapezoidal rule to the integral of 5cos(x) from 0 to π, we have h = (π - 0)/5 = π/5. Evaluating the function at the endpoints and the points within the subintervals and substituting them into the trapezoidal rule formula, we obtain the approximation of approximately 7.42.Simpson's rule is another numerical method used to approximate definite integrals, particularly with smooth functions.

With n = 4, the interval [0, π/2] is divided into 4 subintervals of equal width. The formula for Simpson's rule is given by h/3 * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)].Applying Simpson's rule to the integral of cos(x) from 0 to π/2, we have h = (π/2 - 0)/4 = π/8. Evaluating the function at the endpoints and the points within the subintervals and substituting them into the Simpson's rule formula, we obtain the approximation of approximately 1.02.

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To factor quadratics with leading coefficients other than 1, you can follow these steps:

Check your factoring by expanding the factored form to see if it simplifies back to the original quadratic equation.

Remember that factoring quadratics with leading coefficients other than 1 may involve more complex algebraic techniques, and in some cases, the quadratic equation may not factor easily. In such cases, you can resort to using the quadratic formula to find the roots of the equation.

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Answer: x = 30

Step-by-step explanation:

Answer:

Hi Wdym m=by 100% of will give me 93

Step-by-step explanation:

cos θ = 3/5

sin θ = -4/5

Let's plot the given point on the coordinate plane below: We have the point (-3, 4) in Quadrant IV. Let θ be the angle in the standard position whose terminal side contains the given point. Let's label the hypotenuse as r.

Using the Pythagorean Theorem, we can find r:r² = 3² + 4²r² = 9 + 16r² = 25r = √25r = 5

Now we have:r = 5cos θ = adjacent / hypotenuse cos θ = 3/5sin θ = opposite / hypotenuse sin θ = -4/5Therefore,

cos θ = 3/5

sin θ = -4/5

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Answer:

z+9(8−x)+8(z+8)=

z+72-9x+8z+64=

9z-9x+136

(3−10)^2−(5−12)^2     parenthesis first

(-7)²-(-7)²=

49-(49)=49-49=0

2^2⋅4^3      exponent first

2*2*4*4*4

4*64=256  

A transformation in mathematics is a process of manipulating a polygon.

A transformation in maths is nothing but a process that manipulates a polygon or any other two-dimensional object on a plane or coordinate system. A transformations describe how two-dimensional figures move around a coordinate system.

The original shape of the object before any transformation is called the Pre-Image and the final shape and position of the object is called as the Image.

In mathematics, there are five different transformations:

Dilation, Reflection, Rotation, Shear and Translation

Therefore, transformation is a process of manipulating a polygon.

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Answer: c=−14/5

Step-by-step explanation: Hope this help

Answer:

c = -2 4/5

Step-by-step explanation:

5c + 4 = 2(0 - 5)

Parentheses first

5c + 4 = 2(-5)

5c +4 = -10

Subtract 4 from each side

5c+4-4 = -10-4

5c = -14

Divide by 5

5c/5 = -14/5

c = -14/5

c = -2 4/5

Answer:

64.86

Step-by-step explanation:

Given: (-77.92) + (-8.39) + 59.4 - (-91.77)

The + and - will become -, and the - and - will become +:

-77.92 - 8.39 + 59.4 + 91.77

Finally, calculate:

-86.31 + 151.17

= 64.86

Answer:

36 degrees for each angle.

Step-by-step explanation:

If you divide 180 by 5 (for each point in the star) you get 36 degrees.

The mean age of a group of 5 students is 20.3 years, and the median age is 19 years. If the youngest person is replaced by an even younger individual.

When the youngest person leaves, the mean age decreases. This is because the mean is calculated by dividing the sum of all ages by the total number of individuals. As the youngest person had an age below the mean, their departure reduces the overall sum of ages and consequently lowers the mean.

Similarly, the median age also decreases when the youngest person is replaced by someone younger. The median is the middle value in a sorted list of ages. By replacing the youngest person with an even younger individual, the new age inserted will be smaller than the previous youngest age, which was the median. This causes the median age to decrease.

In conclusion, if the youngest person in a group of 5 students is replaced by an even younger individual, both the mean and median ages will decrease. The departure of the youngest member lowers the overall sum of ages, impacting the mean, while the insertion of a younger age affects the position of the median in the sorted list.

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The area of given polygon is 44 square units from the given graph.

What is Polygon ?

A polygon is a 2-dimensional geometric shape that is formed by joining a finite number of straight line segments to form a closed shape.

The area of each triangle can be found by using the formula:

Area = (base * height)/2

Triangle 1: Base = 4 units, Height = 5 units

Area of Triangle 1 = (4*5)/2 = 10 square units

Triangle 2: Base = 8 units, Height = 3 units

Area of Triangle 2 = (8*3)/2 = 12 square units

The area of the trapezoid can be found by using the formula:

Area = (Sum of parallel sides * Height)/2

Trapezoid: Height = 4 units, Parallel side 1 = 3 units, Parallel side 2 = 7 units

Area of Trapezoid = ((3+7)*4)/2 = 20 square units

Therefore, the total area of the polygon is:

Total Area = Area of Triangle 1 + Area of Triangle 2 + Area of Trapezoid

Total Area = 10 + 14 + 20 = 44 square units

Hence, The area of given polygon is 44 square units from the given graph.

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Answer: if I am not right I am sorry but think it is $0.25

Step-by-step explanation:

Answer:

a^2+b^2=c^2

Step-by-step explanation:

c is the x the side u want to find. usually a and b is given to you

also ^2 this means squared

A symbol that completes the inequality 6x - 3y ___ -12 is: C. ≥.

In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

Next, we would evaluate the inequality by using specific ordered pairs (x, y) as follows;

(0, 0)

6(0) - 3(0) ? -12

0 ≥ -12

(1, 2)

6(1) - 3(2) ? -12

0 ≥ -12

(-1, 2)

6(-1) - 3(2) ? -12

-12 ≥ -12

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The average speeds are 78 km/h and 84 km/h respectively

An equation is an expression that shows the relationship between numbers and variables using mathematical operations like exponents, addition, subtraction, multiplication and division.

The equation for speed is:

Speed = distance / time

a) James drove for 1.5 hours at an average speed of x km/h and then 2.5 hours at an average speed of y km/h.

For distance 1:

x = distance 1 / 1.5

distance 1 = 1.5x

For distance 2:

y = distance 2 / 2.5

distance 2 = 2.5y

Total distance = distance 1 + distance 2

1.5x + 2.5y = 327

multiply through by 2:

3x + 5y = 654    (1)

b) Ryan drove for 4 hours at an average speed of x km/h and then 6 hours at an average speed of y km/h. He drove a total distance of 816 km

For distance 1:

x = distance 1 / 4

distance 1 = 4x

For distance 2:

y = distance 2 / 6

distance 2 = 6y

Total distance = distance 1 + distance 2

4x + 6y = 816     (2)

c) Solving equations 1 and 2 simultaneously:

x = 78; y = 84

The value of x and y are 78 and 84 respectively

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Answer:  $19968

Step-by-step explanation: Assuming the electric technician works 52 weeks a year with an average of of $40 of paid work, the electrician would make about $19,968 or close to $20,000 a year.

The probability that a sample of 40 cans will have a mean amount of at least 15.4 oz can be determined using the central limit theorem and the properties of the normal distribution.

According to the central limit theorem, when sampling from a population with any distribution, as the sample size increases, the distribution of sample means approaches a normal distribution. In this case, we are interested in the mean amount of the sample of 40 cans.

To calculate the probability, we need to standardize the sample mean using the z-score formula: z = (x - μ) / (σ / √n), where x is the desired mean (15.4 oz), μ is the population mean (15.00 oz), σ is the population standard deviation (0.9 oz), and n is the sample size (40).

Calculating the z-score for 15.4 oz, we have: z = (15.4 - 15.00) / (0.9 / √40) ≈ 3.95.

We can then use a standard normal distribution table or statistical software to find the probability associated with a z-score of 3.95. This probability represents the area under the normal curve to the right of 15.4 oz. The probability is very small, close to 0, indicating that the chance of obtaining a sample mean of at least 15.4 oz from the given population is extremely low.

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