find the area of a triangle point

The equation of the tangent plane is given by:
20736(x - 3) + 12288(y - 4) + 1(z - (-62188)) = 0

To find the equation of the tangent plane to the surface x⁴y⁴ + z - 20 = 0 at the point (3, 4, z), we first need to find the partial derivatives with respect to x, y, and z.

∂f/∂x = 4x³y⁴
∂f/∂y = 4x⁴y³
∂f/∂z = 1

Now, we evaluate the partial derivatives at the given point (3, 4, z):

∂f/∂x(3, 4, z) = 4(3³)(4⁴) = 20736
∂f/∂y(3, 4, z) = 4(3⁴)(4³) = 12288
∂f/∂z(3, 4, z) = 1

Next, we find the value of z by substituting x = 3 and y = 4 in the equation:

(3⁴)(4⁴) + z - 20 = 0
z = 20 - (3⁴)(4⁴) = 20 - 62208 = -62188

The point on the surface is (3, 4, -62188). The equation of the tangent plane is given by:

20736(x - 3) + 12288(y - 4) + 1(z - (-62188)) = 0

This simplifies to:

20736x + 12288y + z = 1885580

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Triangle R S P is similar to triangle Z X Y

Triangle S R P is similar to triangle X Z Y

Triangle R P S is similar to triangle Z Y X

Similar triangles, as the name suggests, are two or more regular polygons that share a common form, yet vary in scale. Primarily, this is due to the fact that each shape's corresponding angles are congruent and their matching sides are proportionate.

Hence, if one were to expand or reduce one of the given triangles with a particular factor, it could be properly aligned and matched up with the other triangle. Such characteristics of similar triangles render them to be greatly beneficial in numerous mathematical and geometric undertakings.

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To find the percentage of plants that are exactly 13cm tall using a proportion, you need to first identify the number of plants that are 13cm tall and the total number of plants. Based on your question, let's assume that 2 out of 9 plants are exactly 13cm tall.

Now, set up a proportion with the given information:

(number of plants 13cm tall) / (total number of plants) = (x) / (100)

In this case, the proportion is:
2/9 = x/100

To solve for x, cross-multiply:
2 * 100 = 9 * x
200 = 9x

Now, divide both sides by 9:
x = 200 / 9
x ≈ 22.22
So, approximately 22.22% of the plants in Dorothy's garden are exactly 13cm tall.

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The diameter of the circle whose circumference equals 172.7 cm is approximately 55 cm, as found using the formula c = πd with the given value of π as 3.14. The answer is option A.

The formula for the circumference of a circle is c = πd, where c is the circumference and d is the diameter. We are given that the circumference of the circle is 172.7 cm, and π is approximately 3.14. So we can solve for the diameter as

d = c/π = 172.7/3.14 ≈ 55

Therefore, the diameter of the circle is approximately 55 cm.

The correct answer is option A: 55 cm.

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The solution to the system is x = 94/9 and y = 22/9.

There is a unique solution, we classify the system as consistent and independent.

Part 1: Solve the system using linear combination or substitution. Show all work. (4 points)
System: 3x - 12y = 2, y = x - 8
Part 2: Classify the system as consistent independent, inconsistent, or coincident. (2 points)

Part 1: Let's solve the system using substitution:
Since y = x - 8, we can substitute this expression for y in the first equation:
3x - 12(x - 8) = 2

Now, we'll solve for x:
3x - 12x + 96 = 2
-9x + 96 = 2
-9x = -94
x = 94/9

Now that we have the value of x, we can substitute it back into y = x - 8 to find the value of y:
y = (94/9) - 8
y = (94 - 72)/9
y = 22/9

So, the solution to the system is x = 94/9 and y = 22/9.

Part 2: Since there is a unique solution, we classify the system as consistent and independent.

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There is no convincing evidence that the proportion of Devon's serves that are good is more than 72%. The data collected does not provide sufficient evidence to support Devon's claim that he has a higher proportion of good serves than what his coach stated.

Based on Devon's hypotheses, H₀ states that p = 72%, while Hₐ states that p > 72%, where p represents the true proportion of Devon's good serves. To test this, 50 of his serves are randomly selected, and 42 are good. A simulation is conducted with 100 trials, resulting in an estimated P-value of 0.06. The significance level (α) is set at 0.05.

In this case, the P-value (0.06) is greater than the significance level (0.05). According to the rules of hypothesis testing, we should fail to reject the null hypothesis (H₀) when the P-value is greater than the significance level. Therefore, Devon should fail to reject H₀.

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The estimate for f(3.1, -3.01) using the linear approximation is approximately -54.28.

a. To find the linear approximation of f(x,y) at the point (3,-3), we need to find the gradient of the function at that point.

∇f(x,y) = [-8x, 4y]

So, at the point (3,-3), the gradient is ∇f(3,-3) = [-24, -12].

The linear approximation is given by:

L(x,y) = f(3,-3) + ∇f(3,-3)·(x-3,y+3)

Plugging in the values, we get:

L(x,y) = -4(3)^2 + 2(-3)^2 - 24(x-3) - 12(y+3)

Simplifying, we get:

L(x,y) = -12x - 4y - 36

b. To estimate f(3.1, -3.01) using the linear approximation, we plug in the values into the equation we found in part (a):

L(3.1, -3.01) = -12(3.1) - 4(-3.01) - 36

L(3.1, -3.01) ≈ -54.28

Therefore, the estimate for f(3.1, -3.01) using the linear approximation is approximately -54.28.

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Therefore, the solutions of the equation in the interval [0, 2) are: x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees). Note that the solutions are listed in ascending order.

The equation is:

sec(x) = 2√3/3

First, we can find the values of x for which sec(x) = 2√3/3. Recall that sec(x) = 1/cos(x), so we have:

1/cos(x) = 2√3/3

Multiplying both sides by cos(x), we get:

1 = 2√3/3 cos(x)

cos(x) = 3/(2√3) = √3/2

Now, we can use the unit circle to find the solutions of the equation in the interval [0, 2).

cos(x) = √3/2 when x is π/6 or 11π/6 (in radians), or 30° or 330° (in degrees), since these are the angles in the unit circle where the x-coordinate is √3/2.

However, we need to make sure that these solutions are in the interval [0, 2). Since the period of sec(x) is 2π, we can add or subtract 2π to any solution to get another solution. Therefore, we need to find the solutions in the interval [0, 2π) that correspond to the solutions we found above.

π/6 is already in the interval [0, 2π), so it is a solution in the interval [0, 2). To find the other solution in the interval [0, 2), we can add 2π to 11π/6:

11π/6 + 2π = 23π/6

23π/6 is not in the interval [0, 2), so we need to subtract 2π instead:

11π/6 - 2π = -π/6

-π/6 is in the interval [0, 2), so it is also a solution in the interval [0, 2).

Therefore, the solutions of the equation in the interval [0, 2) are:

x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees)

Note that the solutions are listed in ascending order.

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The management fee Mike must pay this year is $84.

To calculate the management fee Mathew must pay this year, we need to find 0.50 percent of $16,800.

First, let's convert the percentage to a decimal by dividing it by 100. So, 0.50 percent is equal to 0.50/100 = 0.005.

Now, multiply the total investment amount by the management fee rate:

Management fee = $16,800 x 0.005 = $84.

Therefore, Mathew must pay $84.00 as the management fee this year for his investment in the New Colony Pacific Region mutual fund (rounded to 2 decimal places).

In summary, management fees are a percentage of the total asset value invested in a mutual fund, which goes toward compensating the fund managers for their services. In this case, Mathew's management fee is 0.50 percent, which equals $84.00 for the year.

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If Mrs. Smith's estimation only accounted for a 4-week month, it would be incorrect. However, if she factored in the additional 10 days, then her estimation of serving 40,000 people soup with 1 cup per person over 2 weeks is correct.

Soup kitchens are community-based organizations that provide meals, primarily soup, to people in need. These facilities are often run by volunteers and rely heavily on donations from local businesses and individuals to provide food for those who cannot afford it.

Mrs. Smith, the soup kitchen chef, estimates that they will serve 40,000 people soup, giving each person 1 cup of soup over 2 weeks, and with the month having 5 weeks.

To determine if Mrs. Smith's estimation is correct, we need to do some calculations. One cup of soup per person over two weeks means that each person will receive 2 cups of soup in total. Therefore, to serve 40,000 people with 2 cups of soup each, the soup kitchen would need to provide a total of 80,000 cups of soup.

With the month having 5 weeks, it means that there are 10 days extra in the month compared to a standard 4-week month. Therefore, the soup kitchen will need to serve an additional 1/5 of the total amount of soup to cover the additional 10 days.

So, to determine if Mrs. Smith's estimation is correct, we can multiply the total cups of soup needed (80,000) by 1/5, which equals 16,000 cups. We then add this to the original total, which gives us 96,000 cups of soup needed for the month.

In conclusion, if Mrs. Smith's estimation only accounted for a 4-week month, it would be incorrect. However, if she factored in the additional 10 days, then her estimation of serving 40,000 people soup with 1 cup per person over 2 weeks is correct.

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The 95% confidence interval for the true proportion of defective chips is between 0.043 and 0.197.

To calculate the 95% confidence interval for the true proportion of defective computer chips, the quality control specialist would use the formula:

proportion +/- z ×√(proportion x (1-proportion)/sample size)

In this case, the proportion of defective chips is 12/100 or 0.12. The sample size is 100. To find the value of z for a 95% confidence level, we look at a standard normal distribution table or use a calculator and find that it is 1.96.

So the calculation for the confidence interval would be:

0.12 +/- 1.96 × √(0.12 × (1-0.12)/100)

Simplifying this gives us:

0.12 +/- 0.077

This means that if we repeated this sampling process many times, we would expect the true proportion of defective chips to fall within this interval 95% of the time.

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Answer:

Rahul:

[tex]53000( {1.02875}^{7} ) = 64631.59[/tex]

Layla:

[tex]53000 {e}^{.0225 \times 7} = 62040.78[/tex]

$64,631.59 - $62,040.78 = $2,590.81

After 7 years, Rahul's account will have $2,591 more than Layla's account.

The probability that a randomly chosen part has either a major flaw or a minor flaw is 22% or 0.22.

To find the probability that a randomly chosen cast aluminum part has a flaw (major or minor), we can simply add the percentages of parts with minor flaws and major flaws together.

From the given information, 20% of the parts had minor flaws and 2% had major flaws. When we add these percentages together, we get:

20% (minor flaws) + 2% (major flaws) = 22%

Thus, there is a 22% probability that a randomly chosen part has a flaw, either major or minor. Rounded to two decimal places, this would be written as 0.22.

In summary, by considering the percentages of parts with minor and major flaws, we can determine the overall probability of selecting a flawed part. In this case, the probability is 22% or 0.22 when rounded to two decimal places.

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At the end of 5 years, Sam will have paid $1,900 in interest to the bank. This calculation assumes that the interest rate remains constant and is applied on a simple basis, rather than being compounded over the 5-year period.

Sam makes a cash deposit of $8,000 on a used car and is charged an interest rate of 4.75% per year by the bank. To calculate the interest he will pay at the end of 5 years, we can use the formula for simple interest, which is I = P × R × T, where I is the interest, P is the principal (the initial deposit), R is the interest rate, and T is the time in years.

In this case, P = $8,000, R = 4.75% (or 0.0475 in decimal form), and T = 5 years. Plugging these values into the formula, we get:

I = $8,000 × 0.0475 × 5

I = $1,900

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Sam will pay $1,900 in interest to the bank at the end of 5 years.

The interest rate is 4.75% per year, which can be written as 0.0475 as a decimal. The amount of interest that Sam will pay after 5 years can be calculated using the simple interest formula:

Interest = Principal x Rate x Time

where Principal is the initial deposit, Rate is the interest rate, and Time is the length of time the interest is charged for.

In this case, the Principal is $8,000, the Rate is 0.0475, and the Time is 5 years. Plugging these values into the formula, we get:

Interest = $8,000 x 0.0475 x 5

Interest = $1,900

Therefore, Sam will pay $1,900 in interest to the bank at the end of 5 years.

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Answer:

x=50

Step-by-step explanation:

1. multiply the numbers

48x25=24x

1200=24x

2. Divide both sides by the same factor

1200/24 = 24/24x

simplify the expression

x=50

Answer:

Step-by-step explanation:

Assuming that your salary is $1, your savings after each year would be:

End of year 1: $1 x 1.015 = $1.015

End of year 2: $1.015 x 1.015 = $1.03023

End of year 3: $1.03023 x 1.015 = $1.04586

End of year 4: $1.04586 x 1.015 = $1.06186

Therefore, after 4 years of saving your entire salary with a 1.5% increase each year, you would have approximately $1.06.

The length of OP to the nearest tenth of a foot is approximately 41.5 feet

To find the length of OP, we can use the Pythagorean theorem since we have a right triangle.

OP^2 = PN^2 - ON^2

First, we need to find ON using the trigonometric ratio of tangent.

tan(39) = ON/PN

ON = PN * tan(39)
ON = 72 * tan(39)
ON ≈ 53.4 feet

Now we can plug in our values:

OP^2 = 72^2 - 53.4^2
OP^2 ≈ 1720.84
OP ≈ 41.5 feet (rounded to the nearest tenth of a foot)

Therefore, the length of OP to the nearest tenth of a foot is approximately 41.5 feet.

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The calories she burns per minute is 17 cal

We are given that the total calories that are burnt in 15 minutes and 20 minutes are 255 and 340 respectively.

We can use the equation

total calorie burnt = time (in minutes) * calorie burnt in one minute

here we know the total calorie that is burnt and the time, we can substitute the calorie that is burnt in a single minute with 'n'.

we can say that :

255 = 15 * x

x=255/15

x= 17.

The total calorie burn per minute is 17.

now for the verification, we know that if the total calorie burn per minute is 17 it should satisfy both the equation.

So, 340 = 20*x

340=20* 17

340 = 340

Thus it satisfies both equations.

Hence the calorie burnt per minute = 17

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The lid's area can be calculated using the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.

So, for Camille's can of soup, the lid's area would be:

A = 3.14 x (3 inches)^2
A = 3.14 x 9 square inches
A = 28.26 square inches

Therefore, the lid's area is 28.26 square inches.

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Domain restrictions on the given rational equation is x = 3, x = -1 , x = -4

The rational equation is =  [tex]\frac{x}{x+4} + \frac{12}{x^{2} +5x+4} =\frac{8x}{5x-15}[/tex]

Solving each denominator to find out about domain restriction

Putting each value equal to zero

x+4 = 0

x = -4

Here domain restriction is x = -4

x²+5x+4 = 0

x² + 4x + x+ 4 = 0

x(x+4) + 1(x+4) = 0

(x+1)(x+4) = 0

x+1 = 0 and x+4 = 0

x = -1 and x = -4

Here domain restriction is x = - 1 and x =-4

5x-15 = 0

5(x-3) =0

x=3

Here domain restriction is x = 3

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Question is incomplete complete question is :

Find any domain restrictions on the given rational equation:

select all that apply.

o a. x = 0

o b. x= 3

o c. x= -1

d. x= -4

The area of the shaded region in term of π is 6.22π cm²

A sector is a region or space bounded by two radii and an arc. There are two types of sector, the major sector and the minor sector. The of the sector is determined by the angle formed the two radii.

Area of sector = tetha/ 360 × πr²

where r is the radius of the circle.

= 140/360 × π × 4²

= 2240π/360

Area = 6.22 πcm²

Therefore the area of the shaded part is 6.22π cm²

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Answer:

Step-by-step explanation:

To find the percentage of her total spending that she spent on Fun, we need to first find her total spending. We add up the amounts she spent in each category:

\begin{align*}

\text{Total spending} &= \text{Rent} + \text{Food} + \text{Fun} + \text{Other} \\

&= 1200 + 500 + 300 + 200 \\

&= 2200

\end{align*}

So Kara spent a total of $2200 this month.

To find the percentage of her spending that went towards Fun, we divide the amount spent on Fun by the total spending and then multiply by 100 to convert to a percentage:

300/2200 x 100 ≈ 13.6%

So Kara spent approximately 14% of her total spending on Fun.

To find the probability of p(m) given the information provided, we can use the formula:
p(m) = p(m cap c') + p(m cap c)
where c' represents the complement of c, or everything that is not c.

First, we need to find the probability of c' by using the formula:
p(c') = 1 - p(c)
p(c') = 1 - 0.048

p(c') = 0.952
Next, we can find the probability of p(m cap c') by using the formula:
p(m cap c') = p(m) - p(m cap c)

p(m cap c') = p(m cup c) - p(c)
p(m cap c') = 0.524 - 0.048
p(m cap c') = 0.476

Finally, we can substitute these values into the formula for p(m) and solve:
p(m) = 0.476 + 0.044
p(m) = 0.52

Therefore, the indicated probability of p(m) is 0.52.

In simpler terms, p(m) is the probability of event m occurring. To find this probability, we first need to find the probability of event c not occurring, or c'. Then, we can use this information to find the probability of event m occurring but c not occurring, or m cap c'.

Finally, we add this probability to the probability of event m occurring and c occurring, or m cap c, to get the overall probability of event m occurring, or p(m). In this case, the indicated probability of p(m) is 0.52.

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The value of the book after 11 years is $101.38. Therefore, the correct option is C.

Find the value of the book after 11 years with an initial value of $258 and a decrease rate of 8% per year as follows.

1. Convert the percentage decrease to a decimal by dividing it by 100:

8% / 100 = 0.08

2. Subtract the decimal from 1 to represent the remaining value each year:

1 - 0.08 = 0.92

3. Raise the remaining value (0.92) to the power of the number of years (11):

0.92^11 ≈ 0.39197

4. Multiply the initial value of the book ($258) by the calculated remaining value (0.39197):

$258 × 0.39197 ≈ $101.07

Therefore, after 11 years, the value of the book is approximately $101.07, which is closest to option C, $101.38.

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The sum of money shared by both Aziz and Ahmad is $80.

To find the sum of money shared by Aziz and Ahmad, we'll use the given ratio and the difference between their shares.

1. We are given that Aziz and Ahmad share the money in the ratio 3:7. Let's represent Aziz's share as 3x and Ahmad's share as 7x.
2. It's mentioned that Aziz received $32 less than Ahmad. So, we can write an equation as follows: 7x - 3x = $32.
3. Simplify the equation: 4x = $32.
4. Solve for x: x = $32 / 4, x = $8.
5. Now, we can find the shares of Aziz and Ahmad. Aziz's share: 3x = 3 * $8 = $24. Ahmad's share: 7x = 7 * $8 = $56.
6. To find the total sum of money shared, add both shares: $24 + $56 = $80.

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Answer:

y = 2.25

Step-by-step explanation:

The solutions are the points of intersection of the 2 graphs.

(a) To find the slope of the curve where it meets the right-hand pole, we need to differentiate the given equation with respect to x. y = 17cosh(x/17) - 12.

Using the chain rule, we get dy/dx = (17/17)sinh(x/17) = sinh(x/17). Therefore, at x = 7, the slope of the curve is sinh(7/17) ≈ 0.6968.

(b) To find the angle theta between the line and the pole, we can use trigonometry. The slope of the curve at the right-hand pole is the same as the tangent of the angle theta.

Therefore, tan(theta) = 0.6968. Taking the inverse tangent of both sides, we get theta = arctan(0.6968) ≈ 34.33 degrees. Therefore, the acute angle formed by the right pole and the hanging line is approximately 34.33 degrees.

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Surface area of the box is 304 square inches

Two different methods:

Method 1: Sum of the parts

Method 2: General formula for the Surface Area of a box

Method 1: Sum of the parts

For a box, there are 6 sides, all of which are rectangles:

Each of the above pairs has the same area.

The general formula for the area of a rectangle is [tex]A_{rectangle}=length*width[/tex]

As we look at different rectangles, the length of one rectangle may be considered the "width" of another rectangle, and that's okay as we calculate things separately.  (We'll examine how to calculate everything at once in Method 2).

The area for the front/back side is 8in * 10in = 80 in^2

[tex]A_{front}=A_{back}=80~in^2[/tex]

The area for the left/right side is 4in * 8in = 32 in^2

[tex]A_{left}=A_{right}=32~in^2[/tex]

The area for the top/bottom side is 4in * 10in = 40 in^2

[tex]A_{top}=A_{bottom}=40~in^2[/tex]

So, the total surface area is

[tex]A_{Surface~Area} = A_{front} + A_{back} + A_{left} + A_{right} + A_{top} + A_{bottom}[/tex]

[tex]A_{Surface~Area} = (80in^2) + (80in^2) + (32in^2) + (32in^2) + (40in^2) + (40in^2)[/tex]

[tex]A_{Surface~Area} = 304~in^2[/tex]

Method 2: General formula for the Surface Area of a box

There is a formula for the surface area of a box:[tex]A_{Surface~Area~of~a~box} = 2(length*width + width*height + height*length)[/tex]

This formula calculates the area of one of each of the matching sides from the side pairs discussed in Method 1, adds those areas together (giving 3 of the sides), and doubles the result (bringing in the area for the matching missing 3 sides).

For clarity, let's decide that the "10 in" is the width, the "8 in" is the height, and the left over "4 in" is the length.

[tex]A_{Surface~Area~of~the~box} = 2((4in)(10in) + (10in)(8in) + (8in)(4in))[/tex]

[tex]A_{Surface~Area~of~the~box} = 2(40in^2 + 80in^2 + 32in^2)[/tex]

[tex]A_{Surface~Area~of~the~box} = 2(152in^2)[/tex]

[tex]A_{Surface~Area~of~the~box} = 304in^2[/tex]

Bank B is offering a lower interest rate and will result in a lower total cost over the 20-year period. Even though the monthly payment is slightly lower with Bank B, Trevor should choose Bank B because he will save money in the long run due to the lower interest rate.

To compare the two mortgage options, Trevor needs to consider both the interest rate and the monthly payment amount.

Bank A offers a 5% interest rate with a monthly payment of $791.95. The total amount he will pay over 20 years is:

$791.95 x 12 months/year x 20 years = $190,068

Bank B offers a 4.75% interest rate with a monthly payment of $775.47. The total amount he will pay over 20 years is:

$775.47 x 12 months/year x 20 years = $186,113.60

So, in this case, Bank B is offering a lower interest rate and will result in a lower total cost over the 20-year period. Even though the monthly payment is slightly lower with Bank B, Trevor should choose Bank B because he will save money in the long run due to the lower interest rate.

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