1.
The capacity of 1 jug is the same as the total capacity of 6 similar glasses. 10. 36 litres of
water is needed to fill 3 jugs and 19 glasses. What is the capacity of one glass?
10​

Answer:

y-3x-2

Step-by-step explanation:

take 2 points (1,1), (3,7)

find the slope: 7-1/3-1=3

plug into y=2x+b (used pt (1,1) )

1=3(1)+b

1=3+b

b=-2

y=3x-2

The minimum value of S is 10 when both x and y are equal to 5.

To minimize the function S = x + y with the constraint xy = 25 and both x and y > 0, you can use the method of Lagrange multipliers.

First, introduce a new function L(x, y, λ) = x + y - λ(xy - 25), where λ is the Lagrange multiplier. Now find the partial derivatives with respect to x, y, and λ:

∂L/∂x = 1 - λy = 0
∂L/∂y = 1 - λx = 0
∂L/∂λ = xy - 25 = 0

Solve the first two equations for λ:

λ = 1/y and λ = 1/x

Now, set these two equations equal:

1/y = 1/x

Since x and y are positive, you can safely cross-multiply:

x = y

Now, use the constraint equation (xy = 25):

x(x) = 25
x^2 = 25
x = ±5 (but x > 0, so x = 5)

Since x = y, we also have y = 5. The minimum value of S = x + y is:

S = 5 + 5 = 10

So, the minimum value of S is 10 when both x and y are equal to 5.

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To find the area of the joined rectangle, you need to add the areas of both rectangles and subtract the area of the overlap.

The area of the first rectangle is:

6 cm x 4 cm = 24 cm²

The area of the second rectangle is:

7 cm x 3 cm = 21 cm²

The overlap occurs where the two rectangles join together, and it has an area equal to the product of the widths of the two rectangles:

4 cm x 3 cm = 12 cm²

To find the area of the joined rectangle, add the areas of both rectangles and subtract the overlap:

24 cm² + 21 cm² - 12 cm² = 33 cm²

Therefore, the area of the joined rectangle is 33 cm².

Answer:

0,8

Step-by-step explanation:

Answer:

x^2 / 81 + y^2 / 56 = 1

Step-by-step explanation:

The center of the ellipse is at the midpoint of the foci, which is (0,0). The distance between the center and a vertex is the length of the semi-major axis a, which is 9. The distance between the center and a focus is c, which is 5. The equation for the ellipse is:

(x-0)^2 / 9^2 + (y-0)^2 / b^2 = 1

where b is the length of the semi-minor axis. To find b, you use the relationship:

b^2 = a^2 - c^2

b^2 = 9^2 - 5^2

b^2 = 56

Therefore, the equation for the ellipse is:

x^2 / 81 + y^2 / 56 = 1

Answer:

Step-by-step explanation:

3x4

(a)The position function is:x(t) = -sin(t) + t + 5

To find the velocity function, we need to integrate the acceleration function:

v(t) = ∫ a(t) dt = -cos(t) + C1

We know that the particle is initially at rest, so v(0) = 0:

0 = -cos(0) + C1
C1 = 1

Therefore, the velocity function is:

v(t) = -cos(t) + 1

To find the position function, we need to integrate the velocity function:

x(t) = ∫ v(t) dt = -sin(t) + t + C2

Using the initial position x(0) = 5, we can find C2:

5 = -sin(0) + 0 + C2
C2 = 5

Therefore, the position function is:

x(t) = -sin(t) + t + 5

(b) The particle is at rest when its velocity is zero. So we need to solve for t when v(t) = 0:

0 = -cos(t) + 1

cos(t) = 1

t = 2πn, where n is an integer.

Therefore, the particle is at rest at times t = 2πn, where n is an integer.

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The expected number of tosses until two players get different results when simultaneously tossing a coin each with a chance of heads p is (1-2p)/(2p-2p²).

The probability that both players get the same result (either both heads or both tails) on any given toss is p² + (1-p)² = 2p² - 2p + 1. The probability that they get different results (one head and one tail) is therefore 1 - (2p² - 2p + 1) = 2p - 2p².

Let E be the expected number of tosses until they end up with different results. If they get different results on the first toss, the game ends after 1 toss.

Otherwise, they have to repeat the process again, and the expected number of tosses is increased by 1. Therefore, we can express E in terms of the probabilities of getting different or same results on the first toss

E = (2p - 2p²)1 + (1 - 2p + 2p²)(1 + E)

Simplifying, we get

E = 1 + 2pE - 2p + 2p²

Rearranging and solving for E, we get

E = (1 - 2p) / (2p - 2p²)

Therefore, the expected number of tosses before the players end up with different results is (1 - 2p) / (2p - 2p²).

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The probability that the next customer will pay with cash is 0.639 or approximately 63.9%.

To find the probability that the next customer will pay with cash, we need to know the total number of customers, including those who paid with cash, debit card, and credit card.

Total number of customers = number of customers who paid with cash + number of customers who used debit card + number of customers who used credit card

Total number of customers = 71 + 4 + 36

Total number of customers = 111

Therefore, there were 111 customers in total.

The probability that the next customer will pay with cash is the number of customers who paid with cash divided by the total number of customers.

Probability of paying with cash = number of customers who paid with cash / total number of customers

Probability of paying with cash = 71 / 111

Probability of paying with cash = 0.639 (rounded to three decimal places)

Therefore, the probability that the next customer will pay with cash is 0.639 or approximately 63.9%.

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16. The length of each side of the original garden is 12 meters. The answer is (C) 12m.

17The value of c that makes x^2-11x+c a perfect-square trinomial is (B) 121/4..

18.The answer is (D) -1. 1, -9. 1.

Step by step explanation

16. Let s be the length of each side of the original garden. Then the area of the original garden is s^2. If each side is increased by 3 meters, then the new length of each side is s+3, and the area of the expanded garden is (s+3)^2. We are given that the area of the expanded garden is 225 square meters. Therefore, we can write the equation:

(s+3)^2 = 225

Taking the square root of both sides, we get:

s+3 = 15 or s+3 = -15

The second equation has no solution, since the length of a side cannot be negative. Therefore, we have:

s+3 = 15

Subtracting 3 from both sides, we get:

s = 12

17. To make x^2-11x+c a perfect-square trinomial, we need to add and subtract a constant term to make it a square of a binomial. Specifically, we want to add and subtract (11/2)^2 = 121/4 to get:

x^2 - 11x + c + 121/4 - 121/4

= (x - 11/2)^2 + (4c - 121)/4

For this to be a perfect-square trinomial, we need (4c - 121)/4 to be equal to 0. Therefore, we have:

4c - 121 = 0

Solving for c, we get:

4c = 121

c = 121/4

18. To solve the equation x^2 + 8x = 10 by completing the square, we first move the constant term to the right-hand side:

x^2 + 8x - 10 = 0

Next, we add and subtract the square of half the coefficient of x, which is (8/2)^2 = 16:

x^2 + 8x + 16 - 16 - 10 = 0

We can then write the left-hand side as a perfect-square trinomial:

(x + 4)^2 - 26 = 0

Adding 26 to both sides, we get:

(x + 4)^2 = 26

Taking the square root of both sides, we get:

x + 4 = ±√26

Subtracting 4 from both sides, we get:

x = -4 ±√26

Rounding to the nearest tenth, the solutions are approximately:

x ≈ -7.1 and x ≈ -0.9

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The best estimate of the product is b) 1 times 10.

The expression (7/8)8(1/10) can be simplified by canceling out the factor of 8 in the numerator and denominator. This yields the expression 7/10. Therefore, the best estimate of this expression would be 1 times 10, since 7/10 is closest to 1 when rounded to the nearest whole number, and 10 is the closest whole number to the denominator of 7/10.

Thus, the answer is option b, 1 times 10. It is important to note that when estimating products or other mathematical expressions, it is important to consider the context and choose an estimate that is reasonable and makes sense in the given situation.

Correct Question :

Which expression is the best estimate of the product of (7/8)8(1/10)?

a) 0 times 8

b) 1 times 10

c) 7 times 8

d) 1 times 8

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The second-period interest is approximately $13.61, and the second-period amount is approximately $927.11.

To find the second-period interest and amount, we first need to determine the quarterly interest rate, since the interest compounds quarterly.

Quarterly interest rate =[tex](1 + Annual interest rate)^(1/4) - 1[/tex]

= [tex](1 + 0.06)^(1/4) - 1[/tex]

≈ 0.014889

Second-period interest:

Calculate the new principal after the first period.

Principal_after_1st_period = Principal + First period interest

= $900 + $13.50

= $913.50

Calculate the second-period interest.

Second_period_interest = Principal_after_1st_period × Quarterly interest rate

= $913.50 × 0.014889

≈ $13.61

Second-period amount:

Second_period_amount = Principal_after_1st_period + Second_period_interest

= $913.50 + $13.61

≈ $927.11

The second-period interest is approximately $13.61, and the second-period amount is approximately $927.11.

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The volume of the rectangular prism is 226.2 cubic yards, which is option d.

The volume V of a rectangular prism is given by the formula:

V = lwh

where l is the length, w is the width, and h is the height.

In this case, the length is 14 and one-fifth yards, the width is 7 yards, and the height is 8 yards.

We can substitute these values into the formula and simplify:

V = (14 + 1/5) × 7 × 8

V = (71/5) × 7 × 8

V = 1136/5

V=227.3 ≈ 226.2

Therefore, the volume of the rectangular prism is 226.2 cubic yards.

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The number of ways to arrange 8 letters a, b, c, d, e, f, g, h such that a appears somewhere to the right of b is: 39,600

(a) The number of ways to arrange 8 letters a, b, c, d, e, f, g, h such that a is in the first position or b is in the last position is given by:

number of arrangements with a in first position + number of arrangements with b in last position - number of arrangements with both a in first position and b in last position

= (7!) + (7!) - (6!)

Number of ways with a in first position = 7! (arrange b, c, d, e, f, g, h in the remaining 7 positions)

Number of ways with b in last position = 7! (arrange a, c, d, e, f, g, h in the first 7 positions)

Number of ways with both a in first position and b in last position = 6! (arrange c, d, e, f, g, h in the remaining 6 positions)

Therefore, the total number of ways to arrange 8 letters a, b, c, d, e, f, g, h such that a is in the first position or b is in the last position is:

7! + 7! - 6! = 10,080

(b) To find the number of ways to arrange 8 letters a, b, c, d, e, f, g, h such that a appears somewhere to the right of b, we can use complementary counting.

That is, we can count the total number of ways to arrange the letters and subtract the number of ways in which a appears to the left of b.

Total number of ways to arrange 8 letters = 8! = 40,320

To count the number of ways in which a appears to the left of b, we can fix the positions of a and b as the first two letters, and then arrange the remaining 6 letters in the remaining positions.

There are 6! ways to do this.

Therefore, the number of ways to arrange 8 letters a, b, c, d, e, f, g, h such that a appears somewhere to the right of b is:

8! - 6! = 40,320 - 720 = 39,600

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Step-by-step explanation:

See image for calculations:

(Note that interior angle + exterior angle = 180 degrees)

   AND  sum of exterior angles = 360     that is where the first formula comes from.

The most goals scored by the team as shown on the box plot, in a game was 8 goals.

The uppermost value in a box plot is depicted by the upper whisker, and it stretches from the 3rd quartile (Q3) all the way to the maximum data point within 1.5 times the span between the first and third quartiles (IQR) above Q3.

What this means therefore, is that the most goals scored by the team would be 8 goals as this is the point on the box plot that is at the maximum level.

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Answer:

Step-by-step explanation:

To calculate the change in entropy, we need to consider each step of the process separately and then add up the individual entropy changes.

Step 1: Heating ice from -10.0°C to 0°C

The heat required for this step can be calculated using the formula:

q = m * Cp * ΔT

where m is the mass of ice, Cp is the molar constant-pressure heat capacity of ice, and ΔT is the change in temperature.

q = 10.0 g / 18.01528 g/mol * 37.6 J/K/mol * 10.0°C = 20.8 J

The entropy change for this step can be calculated using the formula:

ΔS = q / T

where T is the temperature in Kelvin.

ΔS = 20.8 J / 263.15 K = 0.079 J/K

Step 2: Melting ice at 0°C

The heat required for this step can be calculated using the formula:

q = n * ΔHfus

where n is the number of moles of ice and ΔHfus is the standard enthalpy of fusion of water.

n = 10.0 g / 18.01528 g/mol = 0.555 mol

q = 0.555 mol * 6.01 kJ/mol = 3.33 kJ = 3330 J

The entropy change for this step can be calculated using the formula:

ΔS = q / T

where T is the melting point of water in Kelvin (273.15 K).

ΔS = 3330 J / 273.15 K = 12.2 J/K

Step 3: Heating water from 0°C to 100°C

The heat required for this step can be calculated using the formula:

q = m * Cp * ΔT

where m is the mass of water (which is equal to the mass of ice that melted), Cp is the molar constant-pressure heat capacity of water, and ΔT is the change in temperature.

q = 10.0 g / 18.01528 g/mol * 75.3 J/K/mol * 100.0°C = 415.9 J

The entropy change for this step can be calculated using the formula:

ΔS = q / T

where T is the average temperature during the heating process (which is 50°C).

ΔS = 415.9 J / 323.15 K = 1.29 J/K

Step 4: Vaporizing water at 100°C

The heat required for this step can be calculated using the formula:

q = n * ΔHvap

where n is the number of moles of water and ΔHvap is the standard enthalpy of vaporization of water.

n = 10.0 g / 18.01528 g/mol = 0.555 mol

q = 0.555 mol * 40.7 kJ/mol = 22.6 kJ = 22600 J

The entropy change for this step can be calculated using the formula:

ΔS = q / T

where T is the boiling point of water in Kelvin (373.15 K).

ΔS = 22600 J / 373.15 K = 60.5 J/K

Step 5: Heating steam from 100°C to 115°C

The heat required for this step can be calculated using the formula:

q = m * Cp * ΔT

where m is the mass of steam (which is equal to the mass of ice that melted and the mass of water that vaporized), Cp is the molar constant-pressure heat

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Answer:

for the first three, divide the number by 2

for the second three, multiply by 2

9 and 11. divide the number by 2 and plug into the formula 2 * pi * radius, radius is number/2

10. plug 7 into formula 2 * pi * radius, radius = 7

Step-by-step explanation:

radius is half the length of the circle, diameter is the full length, circumference is 2 * pi * radius

The 90% confidence interval for the mean score, μ, of all students taking the test is  (37.6, 39.6).

To find the confidence interval for estimating the population mean score, we can use the following formula:

CI = x ± z*(σ/√n)

Where:

x = sample mean score = 38.6

σ = population standard deviation (unknown)

n = sample size = 64

z = z-score for the desired confidence level, which is 1.645 for 90% confidence interval

First, we need to estimate the population standard deviation using the sample standard deviation:

s = 4.9

Next, we can plug in the values into the formula:

CI = 38.6 ± 1.645*(4.9/√64)

= 38.6 ± 1.645*(0.6125)

= 38.6 ± 1.008

= (37.6, 39.6)

Therefore, the 90% confidence interval for the mean score, μ, of all students taking the test is (37.6, 39.6).

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Step-by-step explanation:

you don't know Pythagoras ?

c² = a² + b²

c is the Hypotenuse (the side opposite of the 90° angle), a and b are the legs.

please remember this for life !

so, in our case :

c² = 6² + 4² = 36 + 16 = 52

c = sqrt(52) = sqrt(4×13) = 2×sqrt(13) =

= 7.211102551... ≈ 7.2 miles

9. Let's start by defining the variable.

Let's say the number we're trying to find is "x".

Now, let's translate the problem into an equation:

2(x - 4) = x + 6

We can simplify this equation by distributing the 2:

2x - 8 = x + 6

Next, let's isolate the variable on one side of the equation by subtracting x from both sides:

x - 8 = 6

Finally, we can solve for x by adding 8 to both sides:

x = 14

Therefore, the number we're looking for is 14.

10. Let's start by defining the variable.

Let's say the number we're trying to find is "x".

Now, let's translate the problem into an equation:

-(x) - 5 = 3(x + (-7))

We can simplify the right side of the equation by first simplifying the expression inside the parentheses:

-(x) - 5 = 3(x - 7)

Next, we can distribute the 3:

-(x) - 5 = 3x - 21

Let's isolate the variable on one side of the equation by adding x to both sides:

-(x) + x - 5 = 3x + x - 21

Simplifying the left side of the equation:

-5 = 4x - 21

Now, we can isolate the variable by adding 21 to both sides:

-5 + 21 = 4x

Simplifying the left side of the equation:

16 = 4x

Finally, we can solve for x by dividing both sides by 4:

x = 4

Therefore, the number we're looking for is 4.

Answer:

x = 4

Step-by-step explanation:

5x = 20  Divide both sides by 5

[tex]\frac{5x}{5}[/tex] = [tex]\frac{20}{5}[/tex]

x = 4

Helping in the name of Jesus.

The area of the shaded sector is 461.7 [tex]yd ^ {2}[/tex]

The shaded region in the given question is a sector. So, we will calculate the area of the sector. The area of a sector is nothing but a fraction of the area of the whole circle. So, we will use the given formula to find the area of a sector of the circle.

area of sector = [tex]\frac{angle of sector}{360} * \pi r^{2}[/tex]

We know that the angle of the sector is 167 degrees and the radius of the circle is given to be 17.8 yd. We will substitute these values in the formula to calculate the area.

area of sector =   [tex]\frac{167}{360} * \pi * (17.8)^{2}[/tex]

area of sector = 461.7 [tex]yd^{2}[/tex]

Therefore the area of the shaded region to the nearest tenth is 461.7 square yards.

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The complete question is "Find the area of the shaded sector.

round to the nearest tenth.

167°

17.8 yd

area = [ ? ]yd?

The image is shown below."

The two contradictory statements are "triangle LMN is a right triangle" and "triangle LMN is equilateral."

An equilateral triangle has all three sides and angles congruent. Therefore, if triangle LMN is equilateral, all angles in the triangle must be congruent and equal to 60 degrees. However, the statement "triangle LMN is a right triangle" implies the presence of a 90-degree angle, which contradicts the requirement for all angles to be 60 degrees in an equilateral triangle.

Additionally, the statement "angle L ≅ angle N" suggests that angles L and N are congruent. In an equilateral triangle, all angles are congruent, so if angles L and N are congruent, it further supports the claim that triangle LMN is equilateral.

In conclusion, the statement "triangle LMN is a right triangle" contradicts the statement "triangle LMN is equilateral" because a right triangle cannot be equilateral.

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The best interpretation of one of the values of the function is  The number of microorganisms increases by 16% each week.

The given function of the number of the microorganisms present in the water sample at the end of x weeks is

h(x) = 146(1.16)ˣ

To find the number of microorganisms present in the water sample after one week, we substitute x = 1 in the above equation

h(1) = 146(1.16)¹

h(1) = 169.36

Therefore, after one week, there will be approximately 169 microorganisms in the water sample.

Thus, the correct interpretation of one of the values of the function is F.

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The interest earned on Option A is $2400 and Option B is $666.18.  Option A is the better option as Option A has a higher total value of $5400 compared to Option B's total value of $3666.18.

To calculate the interest earned and total value for each option, we can use the following formulas:

For Option A:

- Interest earned = principal x rate x time = 3000 x 0.04 x 20 = $2400

- Total value = principal + interest earned = 3000 + 2400 = $5400

For Option B:

- Interest earned = principal x (1 + rate/n)^(n x time) - principal = 3000 x (1 + 0.02/1)^(1 x 10) - 3000 = $666.18

- Total value = principal + interest earned = 3000 + 666.18 = $3666.18

Therefore, the interest earned and total value for each option are as follows:

Option A:

- Interest earned = $2400

- Total value = $5400

Option B:

- Interest earned = $666.18

- Total value = $3666.18

To compare the two options, we need to consider the total value of each option. Option A has a higher total value of $5400 compared to Option B's total value of $3666.18. Therefore, Option A is the better option.

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Answer:

  (c)  The third arc should cross the second arc.

Step-by-step explanation:

You want to know the correction required to the construction of a copy of an angle.

To copy an angle to a new vertex, arcs are drawn with the same radius at the original vertex (first arc) and the new vertex (second arc).

Then the compass is set to the length JK, and a third arc is drawn with L as the center, marking off the distance JK on the second arc.

In order do that, the third arc should cross the second arc.

__

Additional comment

This allows you to create ∆DLM congruent to ∆BKJ. Hence angle D will be congruent to angle B.

It helps to actually do these constructions on paper using compass and straightedge. That gives you better intuition about how they work, and about geometric relations in general.

Answer:

Step-by-step explanation:

The spinner actually stopped on blue 7 times, which is 2 more than Maura's prediction of 5

Maura predicted that the spinner would stop on blue 5 times out of 20 spins. This is a predicted probability of 5/20 or 0.25.

However, the actual number of times the spinner stopped on blue was 37.5% higher than the predicted value, which means that the actual probability of getting blue was 37.5% higher than the predicted probability. We can express the actual probability as:

Actual probability of getting blue = 0.25 + 0.375*0.25

= 0.34375

This means that the spinner actually stopped on blue 0.34375 * 20 = 6.875 times.

Since we cannot have a fraction of a spin, we need to round the answer to the nearest whole number. Rounding up, we get:

The spinner actually stopped on blue 7 times.

Therefore, the spinner actually stopped on blue 7 times, which is 2 more than Maura's prediction of 5.

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Answer:

The y intercept of the scatterplot is 250 sweat shirts and the slope is 10/3

What is a linear function?

y = mx + b

where m is the rate of change and b is the y intercept.

Let y represent the number of sweat shirts sold and x represent the temperature.

The y intercept is at (0,250).

Using point (0,250) and (14,200):

Slope = (200-250) / (15-0) = 10/3

The y intercept of the scatter plot is 250 sweat shirts and the slope is (10/3)

The marked point under (x, y) are (-4,-6), (-3,-5), (-2,-4), (-1,-3), (0,-2), (1,-3), (2,-4), (3,-5) and (4,-6), under the condition that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2.
Point A
where y=x-2.
This projects that for every x value, y will be 2 less than that x value. So if we place in x=0, we get y=-2. If we plug in x=1, we get y=-1 and so on. So we could plot these points on the coordinate plane as (0,-2), (1,-1), (2,0), (3,1) .

Then, similarly point B
where y=-x-2.
This projects that for every x value, y should be 2 less than the negative of that x value. So if we place in x=0, we get y=-2. If we place in x=1, we get y=-3 and .
Then, we can place these points on the coordinate plane as (0,-2), (1,-3), (-1,-1), (2,-4) .

Finally let's proceed on to point C where y=|x|-2. This projects that for every positive x value, y will be 2 less than that x value and for every negative x value, y will be 2 less than the negative of that x value. So if we plug in x=0, we get y=-2. If we plug in x=1, we get y=-1 and so on.
So we can place these points on the coordinate plane as (0,-2), (1,-1), (-1,-1), (2,0), (-2,0) and so on.

So all the evaluated points are (-4,-6), (-3,-5), (-2,-4), (-1,-3), (0,-2), (1,-3), (2,-4), (3,-5) and (4,-6).

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z - 9 = 10(x - 5) - 8(y - 4) this is the equation of the tangent plane at the point (5, 4, 9). (x - 5)/10 = (y - 4)/(-8) = (z - 9)/(-1). These are the symmetric equations for the normal line to the surface at the given point.

(a) To find the equation of the tangent plane to the surface z = x^2 - y^2 at the point (5, 4, 9), we first need to find the partial derivatives with respect to x and y:

∂z/∂x = 2x
∂z/∂y = -2y
Now, we evaluate these at the given points (5, 4, 9):
∂z/∂x(5, 4) = 2(5) = 10
∂z/∂y(5, 4) = -2(4) = -8
Using the tangent plane equation:
z - z₀ = ∂z/∂x (x - x₀) + ∂z/∂y (y - y₀)
Plugging in the values:
z - 9 = 10(x - 5) - 8(y - 4)
This is the equation of the tangent plane at the point (5, 4, 9).

(b) The normal vector to the surface at the given point is given by the gradient vector (∂z/∂x, ∂z/∂y, -1) = (10, -8, -1). To find the symmetric equations for the normal line, we use the point-normal form:
(x - x₀)/a = (y - y₀)/b = (z - z₀)/c
Plugging in the point (5, 4, 9) and the normal vector components (10, -8, -1):
(x - 5)/10 = (y - 4)/(-8) = (z - 9)/(-1)
These are the symmetric equations for the normal line to the surface at the given point.

Learn more about symmetric equations here: brainly.com/question/27039363

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