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3 years ago

Find two complex numbers that have a sum of 10i, a difference of -4 and a product of -29

Mathematics

1 answer:

HACTEHA [7]3 years ago

5 0

Let these two numbers be $z$ and $w$ . Their sum is $10i$ , their difference is -4, and their product is -29:

$z+w=10i$

$z-w=-4$

$zw=-29$

Add the first two equations together to eliminate $w$ :

$(z+w)+(z-w)=10i-4\implies 2z=-4+10i\implies z=-2+5i$

Then

$w=10i-z\implies w=10i-(-2+5i)=2+5i$

Just to confirm this is correct, check their product:

$(-2+5i)(2+5i)=-4+25i^2=-4-25=-29$

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Estimate the following quantities without using a calculator. Then find a more precise result, using a calculator if necessary.

zubka84 [21]

Explanation:

a. It is so easy to double a number that no approximation is necessary.

31,000 × 200 = 3.1×10⁴ × 2×10² = 6.2×10⁶ exactly

b. 6.2×10³ × 5.2×10⁶ ≈ 6×5×10⁹ = 3×10¹⁰ approximately

The approximation can be refined a bit by taking the ".2" into account:

6.2×10³ × 5.2×10⁶ ≈ (6×5 + .2×(6+5))×10⁹

= 32.2×10⁹ = 3.22×10¹⁰ approximately

Actual product: 3.224×10¹⁰.

For most purposes, the approximation is an adequate approximation, as it it within 10% of the actual value.

c. 9×10⁶ -2.3×10⁴ ≈ 9×10⁶ approximately

A better approximation is to actually subtract an approximation of the smaller number:

9×10⁶ -2.3×10⁴ ≈ (9 - 0.02)×10⁶ ≈ 8.98×10⁶ approximately

The actual value uses all of the digits of the smaller number:

9×10⁶ -2.3×10⁴ ≈ (9 - 0.023)×10⁶ = 8.977×10⁶ exactly

As in part B, either approximation is adequate for most purposes, as the difference from the actual value is less than .3%.

_____

The accuracy required of an approximation, hence the work you expend improving accuracy, should depend on the need in the final application of the number. Often, approximations are used for budget or resource planning purposes where some "slop" is allowed or even expected.

They can also be used in engineering applications, where the error needs to be on the side of more safety (rather than less).

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3 years ago

Ted is constructing an equilateral triangle. He draws a segment with his straightedge and labels the endpoints A and B. He then

gladu [14]

Answer:

Step-by-step explanation:

Keep the same compass opening, place the compass point on B, and draw a circle.

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3 years ago

A sailboat leaves port and sails 12 kilometers west and then 9 kilometers north. The sailboat is now kilometers from port

Luden [163]

Answer: The sailboat is at a distance of 15 km from the port.

Step-by-step explanation: Given that a sail boat leaves port and sails 12 kilometers west and then 9 kilometers north.

We are to find the distance between the sailboat from the port in kilometers.

Since the directions west and north are at right-angles, we can visualize the movement of the sailboat in the form of a right-angled triangle as shown in the attached figure.

The sailboat moves leaves the port at P and reach O after sailing 12 km west. From point O, again it moves towards north 9 km and reach the point S.

PS = ?

Using the Pythagoras theorem, we have from right-angled triangle SOP,

$PS^2=OS^2+OP^2=9^2+12^2=81+144=225\\\\\Rightarrow PS=15~\textup{km}.$