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8 days ago

Homer wants to identify the center and radius of the circle defined by the equation x2 + y2 - 14x +

Mathematics

1 answer:

tester [8.8K]8 days ago

8 0

Homer made an error in step 2

When completing the square, it's necessary to apply the adjustments to both sides of the equation

Solution:

The provided equation for the circle:

$x^2 + y^2 -14x + 2y -25 = 0$

First, let’s determine the center and the radius and identify where the mistake occurred

Step 1:

Group the terms

$x^2 -14x + y^2 + 2y = 25\\\\(x^2 -14x) + (y^2 + 2y) = 25$

Step 2:

$(x^2 -14x + 49) + (y^2 + 2y + 1) = 25 + 49 + 1$

However, Homer’s second step is:

$(x^2 -14x + 49) + (y^2 + 2y + 1) = 25$

Thus, Homer committed an error during this step

When completing the square, the values must be added to both sides of the equation

However, Homer neglected to add 49 and 1 to the right side of the equation

Correct steps are:

$(x^2 -14x + 49) + (y^2 + 2y + 1) = 75\\\\(x-7)^2 + (y+1)^2 = 75$

By comparing with the general equation of a circle,

$(x-h)^2 + (y-k)^2 = r^2$

It follows that the center coordinates are ( 7, -1) and the radius measures 8.66

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Sanjit drives the 45 km distance between York and Leeds he then drives a further 42 km from Leeds to Blackpool

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The average speed for his entire journey from York to Blackpool is about 61.41 km/h.

Here’s a breakdown of how we arrive at this:

$\text{ The average speed}=\frac{\text{Total distance covered}}{\text{Total time taken}}$

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Therefore, the time taken to travel from York to Leeds is 45/54 hours (since Time = Distance/Speed).

Next, the distance from Leeds to Blackpool is 42 km,

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This leads to the total duration for his trip as

$=\frac{45}{54}+\frac{35}{60}=\frac{450+315}{540}=\frac{765}{540}=\frac{17}{12}$ hours.

The cumulative distance covered equals 45 + 42 = 87 km.

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The line width of a tool used for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micromete

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a. 0.82%

b. 71.11%

c. 0.564 micrometer

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To find the z value for each measurement, we must calculate and determine the percentage they correspond to, and the difference will give us the percentage between those two statistics.

The equation for z is:

z = (x - m) / (sd)

Here, x is the value being evaluated, m denotes the mean, and sd represents the standard deviation.

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z = (0.62 - 0.5) / (0.05)

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This translates to 0.9918.

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p (x > 0.62) = 0.0082 = 0.82%

For 0.47 copies, we have:

z = (0.47 - 0.5) / (0.05)

z = -0.6, which equates to 0.2742.

For 0.63 copies:

z = (0.63 - 0.5) / (0.05)

z = -2.6, which yields 0.9953.

Hence, p (0.47 > x > 0.63) = 0.9953 - 0.2742

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p = 0.9 corresponds to z = 1.28.

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