What is the equation of the line containing the point (3,1),(9,3) and (27,9)

Using equation of parabola in vertex form the year in which the average gas mileage for new vehicles sold in Switzerland the greatest is 2016.

The equation of a parabola with vertex (h, k) is given by

y = a(x - h)² + k

Now a researcher found that for the years 2013 to 2019, the equation, y = -0.4(x - 3)² + 42 models the average gas mileage of new vehicles sold in Switzerland, where is the number of years since 2013 and is the average gas mileage, in miles per gallon (mpg).

To determine during what year was the average gas mileage for new vehicles sold in Switzerland the greatest, we notice that the equation is the equation of a parabola in vertex form where (h, k) is the vertex.

Comparing y = a(x - h)² + k with y = -0.4(x - 3)² + 42 we have that

a = -0.4, h = 3 and k = 42

So, the vertex is at (h, k) = (3, 42)

Since a = -0.4 < 0, (3,42) is a maximum point

So, y is maximum when x = 3

Since this is 3 years after 2013 which is 2013 + 3 = 2016.

So, the year in which the average gas mileage for new vehicles sold in Switzerland the greatest is 2016.

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Answer:

  a)  even

  b)  4th differences: -72

  c)  minimum: 0, maximum: 4. This function has 0 real zeros.

  d)  -1377

  e)  -288

Step-by-step explanation:

You want to know a number of the characteristics of the function f(x) = -3x⁴ +6x² -10:

A function is even if f(x) = f(-x). The graph of an even function is symmetrical about the y-axis. An even polynomial function will only have terms of even degree.

The exponents of the terms of f(x) are 4, 2, 0. These are all even, so we can conclude the function is an even function.

We can also evaluate f(-x):

  f(-x) = -3(-x)⁴ +6(-x)² -10 = -3x⁴ +6x² -10 ≡ f(x) . . . . . the function is even

We can look at values of x on either side of x=0. The attachment shows function values and finite differences for x = -3, -2, ..., +3.

The fourth finite differences are constant at -72. (We expect this value to be -3·4!, the leading coefficient times the degree of the polynomial, factorial.)

A 4th-degree polynomial will always have exactly four zeros. They may be complex, rather than real. Complex zeros will come in conjugate pairs, so the number of real zeros may be 0, 2, or 4; a minimum of 0 and a maximum of 4.

This polynomial function has no real zeros. The four complex zeros are approximately ...

  ±1.18864247 ±0.64255033i

The average rate of change on the interval [a, b] is given by ...

  AROC = (f(b) -f(a))/(b -a)

For [a, b] = [2, 7], this is ...

  AROC = (((-3(7²) +6)7² -10) -((-3(2²) +6)2² -10)/(7 -2)

  = ((-147 +6)(49) -(-12 +6)(4)) / 5 = (-6909 +24)/5 = -6885/5 = -1377

The average rate of change on [2, 7] is = -1377.

The derivative of the function is ...

  f'(x) = -3(4x³) +6(2x) = 12x(-x² +1)

  f'(3) = 12·3(-3² +1) = 36(-8) = -288

The instantaneous rate of change at x=3 is -288.

The equation in slope-intercept form for the line with slope (2)/(3) and y-intercept -6 is:

y = (2) / (3)x - 6.

The equation in slope-intercept form for a line is y = mx + b, where m is the slope and b are the y-intercept. Since the slope is (2)/(3) and the y-intercept is -6, we can substitute these values into the equation to get:

y = (2)/(3)x + (-6)

Simplifying this equation gives us:

y = (2)/(3)x - 6

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Answer:

Let's call the first of the three consecutive integers "x".

According to the problem, the third integer (which is the one after the first two) is equal to twice the first increased by five. We can express this algebraically as:

third integer = 2x + 5

Since the three integers are consecutive, the second integer must be one more than the first, and the third must be one more than the second. So, the second integer can be expressed as:

second integer = x + 1

And the third integer is:

third integer = (x + 1) + 1 = x + 2

Now we can set these two expressions for the third integer equal to each other, since they both represent the same value:

2x + 5 = x + 2

Simplifying and solving for x, we get:

x = -3

So the first of the three consecutive integers is -3. The second is one more than the first, which is -3 + 1 = -2. And the third is one more than the second, which is -2 + 1 = -1. Therefore, the three consecutive integers are -3, -2, and -1.

Multiplying or dividing both sides by a positive number leaves the inequality symbol unchanged.

The inequality symbols and > are defined in this pamphlet, along with examples of how to work with expressions containing them.

The following guidelines should be followed when changing or rearranging statements that involve inequalities:

Rule 1: An inequality symbol remains unchanged when the same amount is added to or subtracted from both sides.

Rule 2: Adding or subtracting a positive number from both sides does not change the inequality symbol.

Rule 3: Reversing the inequality by multiplying or dividing both sides by a negative number. It follows that  changes to > and vice versa.

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The monthly mortgage payment for a 30-year mortgage loan of $250,000 with an annual interest rate of 3% is about $1,054.63.

A monthly mortgage payment is the amount of money paid each month to repay a mortgage loan. The payment is typically made up of principal the amount borrowed and interest the cost of borrowing the money and may also include additional amounts for taxes and insurance.

We can use the formula for the monthly mortgage payment, which is:

M = P * r * (1 + r)^n / ((1 + r)^n - 1)

Where

First, we need to convert the annual interest rate to a monthly interest rate:

r = 3% / 12 = 0.0025

Next, we can plug in the values:

M = 250000 * 0.0025 * (1 + 0.0025)^360 / ((1 + 0.0025)^360 - 1)

We can simplify this expression and find that the monthly mortgage payment is approximately $1,054.63.

Therefore, the monthly mortgage payment for a 30-year mortgage loan of $250,000 with an annual interest rate of 3% is about $1,054.63.

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Option 4: 2 feet, 3 feet, and 5 feet – constitutes a right angle since 2² + 3² = 4 + 9 = 13 and 13 = 5², making it.

A triangle is a geometry with three vertices and three sides. The internal angle of the triangle, which really is 180 degrees, is built. The inner triangle angles are implied to sum to 180 degrees. It has the fewest sides of any polygon.

The Pythagorean theorem states that the square of a hypotenuse's length (the side exact reverse the right angle) in a right triangle is the product of a squares of the durations of the remaining two sides. Only the last option—2 feet, 3 feet, and 5 feet—can represent the second derivative of a right triangle because it satisfies this requirement.

Let's check each option:

Option 1: 14 feet, 14 feet, 14 feet - As all three are equal, this doesn't qualify as a right triangle and the Pythagoras theorem cannot be met.

Option 2: 10 feet, 24 feet, and 26 feet - Because 10² + 24² = 100 + 576 = 676, which is equivalent to 26², this is a right triangle. The fact that this option is a multiple of the well-known Polynomial triple (3, 4, and 5) implies that we can scale all of the corresponding sides by a common factor to produce an infinite number of right triangles with all these side lengths. As a result, this option doesn't really represent an original right triangle.

Option 3: 3 ft, 3 ft, 18 ft - This does not constitute a right triangle because the cube of the hypotenuse's length (18² = 324) does not equal the total of the squares of a shorter side (3² + 3² = 18).

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Answer:

2 feet, 3 feet, and 5 feet

Step-by-step explanation:

got it right on my test

Answer:

on what?

Step-by-step explanation:

How do you get rid of an inner bully?

5 Ways to Stop that Inner Bully

Become aware of what you are saying to yourself. ...

Replace this with mindful attention to your feelings. ...

Realize you are not alone in your suffering. ...

Use soothing self-talk. ...

Access Your Wise Mind.

The rate at which the water level is rising when the water is 3m deep is 0.159 m/min.  The rate at which the tip of his shadow is moving when he is 40ft from the pole is 3ft/sec. The volume of a cone is given by V = 1/3πr^2h.

We are given that the base radius is 2m and the height is 4m. We are also given that the rate at which water is being pumped into the tank is 2 m^3/min. We need to find the rate at which the water level is rising when the water is 3m deep.

To find the rate at which the water level is rising, we need to take the derivative of the volume with respect to time. This gives us:

dV/dt = (1/3)π(2r)(dr/dt)(4) + (1/3)π(2^2)(dh/dt)

We know that dV/dt = 2 and r = 2, so we can plug these values into the equation and solve for dh/dt:

2 = (1/3)π(2)(2)(dr/dt)(4) + (1/3)π(2^2)(dh/dt)

Solving for dh/dt gives us:

dh/dt = (6 - 4π(dr/dt))/(4π)

We are given that the water level is 3m deep, so we can plug this value into the equation for the volume of a cone and solve for r:

V = (1/3)πr^2h

3 = (1/3)πr^2(3)

r = √(3/π)

We can now plug this value of r into the equation for dh/dt and solve for dr/dt:

dh/dt = (6 - 4π(√(3/π))(dr/dt))/(4π)

Solving for dr/dt gives us:

dr/dt = (6 - 4π(dh/dt))/(4π√(3/π))

We can now plug this value of dr/dt back into the equation for dh/dt and solve for dh/dt:

dh/dt = (6 - 4π((6 - 4π(dh/dt))/(4π√(3/π))))/(4π)

Solving for dh/dt gives us:

dh/dt = 0.159 m/min

The street light is mounted at the top of a 15ft tall pole and the man is 6ft tall. The man is walking away from the pole with a speed of 5ft/sec along a straight path. We need to find the rate at which the tip of his shadow is moving when he is 40ft from the pole.

We can use the properties of similar triangles to relate the height of the pole, the height of the man, the distance of the man from the pole, and the length of the shadow. Let x be the distance of the man from the pole and y be the length of the shadow. Then we have:

15/x = 6/(x + y)

Cross-multiplying gives us:

15(x + y) = 6x

Simplifying gives us:

9x = 15y

Taking the derivative of both sides with respect to time gives us:

9(dx/dt) = 15(dy/dt)

We are given that dx/dt = 5ft/sec, so we can plug this value into the equation and solve for dy/dt:

9(5) = 15(dy/dt)

Solving for dy/dt gives us:

dy/dt = 3ft/sec

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The value of (x,y) is ( 6, -6) (optionA)

Simultaneous equations are two or more algebraic equations that share variables e.g. x and y . They are called simultaneous equations because the equations are solved at the same time. For example, below are some simultaneous equations: 2x + 4y = 14, 4x − 4y = 4. 6a + b = 18, 4a + b = 14.

6x+4y = 12 equation 1

-6x +6y = -72 equation 2

add equation 1 and 2

10y = - 60

y = -60/10

y = -6

substitute -6 for y in equation 1

6x +4(-6) = 12

6x -24 = 12

6x = 12+24

6x = 36

x = 36/6 = 6

therefore the value of (x,y) = ( 6, -6)

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Answer:

y = (-5/2)x - 3

Step-by-step explanation:

To find an equation of a line that is perpendicular to the given line and passes through the point (0, -3), we need to use the fact that perpendicular lines have opposite reciprocal slopes.

The given line has a slope of 2/5, so the slope of the line perpendicular to it is:

-1 / (2/5) = -5/2

This means that the equation of the perpendicular line has the form:

y = (-5/2)x + b

where b is the y-intercept we want to find.

Since the line passes through the point (0, -3), we can substitute these values into the equation and solve for b:

-3 = (-5/2)(0) + b

b = -3

Therefore, the equation of the line perpendicular to y = 2/5x + 4 with a y-intercept of -3 is:

y = (-5/2)x - 3

The remainder when 7x⁴ - 3x is divided by x - 1 is 4.

knοwn as indeterminates) and cοefficients, which are cοmbined using the οperatiοns οf additiοn, subtractiοn, and multiplicatiοn. A pοlynοmial can have οne οr mοre variables, but each term in the pοlynοmial must have nοn-negative integer expοnents οn the variables. The degree οf a pοlynοmial is the highest pοwer οf its variables with a nοn-zerο cοefficient.

Fοr example, the pοlynοmial 3x² - 2x + 5 has a degree οf 2, with the term 3x² being the highest degree term. The cοefficient οf the term 3x^2 is 3, and the cοefficient οf the term -2x is -2.

Pοlynοmials are used in a variety οf mathematical applicatiοns, including algebra, calculus, and geοmetry. They are used tο represent mathematical functiοns, tο apprοximate cοmplex curves, and tο sοlve equatiοns. Sοme cοmmοn οperatiοns οn pοlynοmials include additiοn, subtractiοn, multiplicatiοn, divisiοn, and factοring.

Tο find the remainder when the pοlynοmial 7x⁴ - 3x is divided by x - 1, we can use pοlynοmial lοng divisiοn οr synthetic divisiοn.

7x³ + 7x² + 7x + 4

x - 1 | 7x⁴ + 0x³ - 3x² + 0x + 0

     - (7x⁴ - 7x³)

           7x³ - 3x²

           - (7x³ - 7x²)

                 4x² + 0x

                 - (4x² - 4x)

                        4x

                        - (4x - 4)

                             4

Therefore, the remainder when 7x⁴ - 3x is divided by x - 1 is 4.

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Answer:

Step-by-step explanation:

A line that is parallel to the first line will have the same slope, so:

m = -3

X1 and y1 are basically the coordinates where the new line intersects, which is x1 = -1, and y1 = 6

Point-slope form:

y - 6 = -3(x - (-1))

y-6 = -3(x+1)

Slope-intercept form:

y - 6 = -3x - 3

y = -3x + 3

Hope this helps!

Answer:

Step-by-step explanation:

(-1,6) + (-3x + 4) = (-4x,10). I don't know if this is really correct but that's all that I really know how and what to do, so I hope I at least kind of helped a little bit.

Yes, the graph represents a function.

A relation is a function if it has only One y-value for each x-value.

The given ordered pairs from the given graph are (-7, 3), (-3, -3), (0,1), (2, 4), (3, -1), (5, -6)

The given graph represents a relation.

Since each value of x has unique y value.

So the given graph represents a function.

Hence, yes the graph represents a function.

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Answer:

p^(2(s-t)^2)/(s+t)

Step-by-step explanation:

We can simplify this expression by using the properties of exponents:

((p^r)/(p^s))^(r+s) ((p^2)/(p^t))^(s+t) ((p^t)/(p^r))^(r+t)

= (p^(r+s-s))^r (p^(2s-2t))^s (p^(t-r+r))^t / (p^(r+s-r))^r (p^(2t-2s))^s (p^(r-t+t))^t

= p^r p^(2s-2t)s p^t / p^r p^(2t-2s)s p^t

= p^r / p^r * (p^(2s-2t))^(s/(s+t)) / (p^(2t-2s))^(s/(s+t))

= p^r / p^r * p^((2s-2t)s/(s+t)) / p^((2t-2s)s/(s+t))

= p^0 * p^(2s^2-2st-2ts+2t^2)/(s+t)

= p^(2s^2-2st-2ts+2t^2)/(s+t)

= p^(2(s-t)^2)/(s+t)

Therefore, ((p^r)/(p^s))^(r+s) ((p^2)/(p^t))^(s+t) ((p^t)/(p^r))^(r+t) simplifies to p^(2(s-t)^2)/(s+t).

4(3)(2)/2 + 2² = 12 + 4 = 16

Answer:

1-B
2-E

3-D

4-A

5-C

Step-by-step explanation:

-4x + 3y = 3

3y = 4y + 3

y = 4/3 y + 1 => slope is 4/3, y-intercept is (0,1)

Equation 1 matches with Letter B

12x - 4y = 8

4y = 12x - 8

y = 3x - 2 => slope is 3, y-intercept is (0,-2)

Equation 2 matches with Letter E

8x + 2y = 16

2y = -8x + 16

y = -4x + 8 => slope is -4, y-intercept is (0,8)

Equation 3 matches with Letter D

-x + 1/3 y = 1/3

1/3 y = x + 1/3

y = 3x + 1 => slope is 3, y-intercept is (0,1)

Equation 4 matches with Letter A

-4x + 3y = -6

3y = 4x = -6

y = 4/3 x - 2 => slope is 4/3, y-intercept is (0,-2)

Equation 5 matches with Letter C

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The correct match of each ordered pair with each function is:

(-9, 74) for y = -8x + 2

(2,-6) for y = -4x + 2

(9, 70) for y = 7x + 7

(-4, 23) for y = -7x - 5

To match each function with the correct ordered pair, we need to substitute the x-values from the ordered pairs into each function and see which one gives the corresponding y-value.

Substitute the x value of (-9, 74) into y = -8x + 2:

y = -8(-9) + 2

y = 74

Substitute the x value of (2,-6) into y = -4x + 2:

y = -4(2) + 2

y = -6

Substitute the x value of (9, 70) into y = 7x + 7:

y = 7(9) + 7

y = 70

Substitute the x value of (-4, 23) into y = -7x - 5:

y = -7(-4) - 5

y = 23

Therefore, the correct matching is:

(-9, 74) for y = -8x + 2

(2,-6) for y = -4x + 2

(9, 70) for y = 7x + 7

(-4, 23) for y = -7x - 5

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The answer of the given question based on the rectangle and a hexagon , the area of the shaded region is approximately 10.51 cm².

Area is  measure of  size of  two-dimensional surface or shape, like  a square, circle, or triangle. It is typically expressed in square units, like square meters (m²) or square feet (ft²).

To find the area of the shaded region in the figure, we need to find the area of the rectangle and the area of the hexagon, and then subtract the area of the hexagon from the area of the rectangle.

The rectangle has a length of 8 cm and a width of 2 cm, so its area is:

A(rectangle) = length x width = 8 cm x 2 cm = 16 cm²

The hexagon has a side length of 2 cm, so we can divide it into 6 equilateral triangles with side length 2 cm. Each of the  triangles has  area of an;

A(triangle) = (sqrt(3)/4) x side² = (sqrt(3)/4) x 2² = (2sqrt(3))/4 = sqrt(3)/2

The area of the hexagon is therefore:

A(hexagon) = 6 x A(triangle) = 6 x sqrt(3)/2 = 3sqrt(3)

A(shaded) = A(rectangle) - A(hexagon) = 16 cm² - 3sqrt(3) cm² ≈ 10.51 cm²

Therefore, the area of the shaded region is approximately 10.51 cm².

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The answer is no, these data do not indicate that the percentage of wheat crop lost to hail in that county is different (either way) from the national mean of 11%.

The sample mean is x = 12.1% and the population mean is μ = 11%. We want to test if there is a significant difference between the sample mean and the population mean. We can use a t-test to compare the means.

The null hypothesis is H0: μ = 11%, and the alternative hypothesis is Ha: μ ≠ 11%.

The t-statistic is calculated as:

t = (x - μ) / (σ / √n)

where x is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size.

Plugging in the values, we get:

t = (12.1 - 11) / (5.0 / √16)
t = 1.1 / (5.0 / 4)
t = 0.88

Using a t-table with degrees of freedom (df) = 16 - 1 = 15 and α = 0.01, we find the critical value to be 2.947. Since the absolute value of the t-statistic (0.88) is less than the critical value (2.947), we fail to reject the null hypothesis. This means that there is not enough evidence to suggest that the percentage of wheat crop lost to hail in that county is different from the national mean of 11%.

Therefore, the answer is no, these data do not indicate that the percentage of wheat crop lost to hail in that county is different (either way) from the national mean of 11%.

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The  interval notation  of x^(2 )-2x-35>0  is (-∞,-5)∪(7,∞).

To solve the quadratic inequality x^(2)-2x-35>0, we first need to find the roots of the quadratic equation x^(2)-2x-35=0. We can do this by factoring the equation:

(x-7)(x+5)=0

The roots of the equation are x=7 and x=-5. Now, we can use these roots to determine the intervals where the inequality is true. We can do this by testing values in each interval:

- For x<-5, let's test x=-6: (-6)^(2)-2(-6)-35=1>0, so the inequality is true in this interval.
- For -57, let's test x=8: (8)^(2)-2(8)-35=29>0, so the inequality is true in this interval.

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polynomial expression of degree 2 with 3 terms.

a. i. \( a^{3}\left(a^{2}+a+1\right) = a^{5}+a^{4}+a^{3} \)


ii. \( \left(3 x^{2}-4 x+3\right)-(4 x-10) = 3 x^{2}-7 x-7 \)


b. i. \( a^{5}+a^{4}+a^{3} \) is a polynomial expression of degree 5 with 3 terms.


ii. \( 3 x^{2}-7 x-7 \) is a polynomial expression of degree 2 with 3 terms.

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Answer: LCM is 72. GCF is 6.

The solution to the compound linear inequality 1.4 ≤ 9.2 - 0.8x ≤ 6.9 are the values of x in the interval [2.9, 9.8].

To solve the compound linear inequality, we need to isolate the variable on one side of the inequality. We can do this by following the same steps as we would when solving a regular equation, but remembering to flip the inequality sign if we multiply or divide by a negative number.

1.4 ≤ 9.2 - 0.8x ≤ 6.9

First, we'll subtract 9.2 from all sides of the inequality:

-7.8 ≤ -0.8x ≤ -2.3

Next, we'll divide all sides by -0.8 to isolate the variable. Remember to flip the inequality signs since we're dividing by a negative number:

9.75 ≥ x ≥ 2.875

Finally, we'll write the solution to the nearest tenth in interval notation:

[2.9, 9.8]

So, the solution to the compound linear inequality are all values of x, which is greater than or equal to 2.9 but less than or equal to 9.8, or x is in the interval [2.9, 9.8].

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Answer:

128

Step-by-step explanation:

Answer: 128

Step-by-step explanation:

Remember the order of operations in this problem:

8² x (2 + 6) / 4

8² x (8) / 4

64 X 8 /4

512 / 4

= 128

Hope this helps!

1) The center of mass of the solid bounded by x = y^2 and the planes x = z, z = 0, and x = 1  the center of mass of the solid is (1/3, 2/15, 1/3). 2) The total charge on the disk is 4/3 coulombs. 3) The center of mass of the triangular region is (2/3, 2/3).


Center of Mass = (∫xyzρdV)/(∫ρdV).

Here, V is the volume of the solid. Since the density is constant, we can pull it out of the integral:

Center of Mass = k*(∫xyzdV)/(∫dV).

We can now use the volume formula for the solid which is V = ∫xyzdxdyz. Plugging this in the above formula, we get:

Center of Mass = k*[(∫x∫ydxdyz)/(∫dxdyz)]

Evaluating the integrals, we get the x coordinate of the center of mass to be (1/3), the y coordinate to be (2/15) and the z coordinate to be (1/3). Thus, the center of mass of the solid is (1/3, 2/15, 1/3).

2. To find the total charge Q on the disk x^2 + y^2 ≤ 1 such that the charge density at any point (x, y) is rho(x, y) = x + y + x^2 + y^2 (in coulombs per square meter), we need to use the following formula:

Q = ∫∫rho(x, y)dxdy

Evaluating the integral, we get Q = (1/3) + (1/3) + (1/3) + (1/3) = 4/3. Thus, the total charge on the disk is 4/3 coulombs.

3. To find the center of mass of the triangular region with vertices (0, 0), (2, 0) and (0, 2) if the density is given by rho(x, y) = 1 + x + 2y, we need to use the following formula:

Center of Mass = (∫xyρdA)/(∫ρdA).

Here, A is the area of the triangle. Evaluating the integral, we get the x coordinate of the center of mass to be (2/3) and the y coordinate to be (2/3). Thus, the center of mass of the triangular region is (2/3, 2/3).

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The missing side in the triangle has a length of 9.899.

In an isosceles triangle, the two sides are equal and the angles opposite to the two equal sides are also equal.

In the figure, ∠RQS = ∠RSQ =45°

Thus, it is an isosceles triangle with sides RQ=RS= 7

What is Pythagoras' theorem?

According to Pythagoras' theorem for a right-angled triangle:

Base² + Height²= Hypotensuse²

In the given figure: Base = 7 and Height = 7,

Thus, QS²= RQ² + RS²

          QS² = 7² + 7²

           QS = √98

                 =9.899    

Learn more about Pythagoras' theorem here:

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Answer:

Jeremiah ate more, by 5 servings more

Step-by-step explanation:

$100 is 4 x number of vegetable servings

100/4 = 25 number of total servings

If Jeremiah ate 15 servings, his brother ate 25-15 =10

Servings Jeremiah 15, brother 10

Answer:

OPTION C

Step-by-step explanation:

There are 3 sides in a triangle. 2 of them are legs, and one of them is the Hypotenuse. "Sin" refers to Opposite/Hypotenuse.

To find A given a sine value, we must use inverse sin. I would suggest using desmos for this, but you need to switch to degrees in the online caluclator.

So the Equation is: [tex]sin^{-1} (0.9659)[/tex]

After plugging that into desmos, we get 74.994 degrees. Because that is not one of the answer, I'm assuming we must round our answer to the nearest whole number. In that case, your answer is 75 degrees, or OPTION C

Answer:

r=7

Step-by-step explanation:

Cylinder Area

= πr² x h

1078 = 22/7 x r² x 7

1078/22 = r²

49=r²

r=7