1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
AveGali [126]
2 years ago
8

A worker was paid a salary of $10,500 in 1985. Each year, a salary increase of 6% of the previous year's salary was awarded. How

much did the worker earn from the beginning of 1985 through the end of 2010?
A $45,064.64
B $256,527.63
C $576,077.38
D $621,142.02
Mathematics
1 answer:
Mazyrski [523]2 years ago
4 0
Note that 6% converted to a decimal number is 6/100=0.06. Also note that 6% of a certain quantity x is 0.06x.

Here is how much the worker earned each year:


In the year 1985 the worker earned <span>$10,500. 

</span>In the year 1986 the worker earned $10,500 + 0.06($10,500). Factorizing $10,500, we can write this sum as:

                                            $10,500(1+0.06).



In the year 1987 the worker earned

$10,500(1+0.06) + 0.06[$10,500(1+0.06)].

Now we can factorize $10,500(1+0.06) and write the earnings as:

$10,500(1+0.06) [1+0.06]=$10,500(1.06)^2.


Similarly we can check that in the year 1987 the worker earned $10,500(1.06)^3, which makes the pattern clear. 


We can count that from the year 1985 to 1987 we had 2+1 salaries, so from 1985 to 2010 there are 2010-1985+1=26 salaries. This means that the total paid salaries are:

10,500+10,500(1.06)^1+10,500(1.06)^2+10,500(1.06)^3...10,500(1.06)^{26}.

Factorizing, we have

=10,500[1+1.06+(1.06)^2+(1.06)^3+...+(1.06)^{26}]=10,500\cdot[1+1.06+(1.06)^2+(1.06)^3+...+(1.06)^{26}]

We recognize the sum as the geometric sum with first term 1 and common ratio 1.06, applying the formula

\sum_{i=1}^{n} a_i= a(\frac{1-r^n}{1-r}) (where a is the first term and r is the common ratio) we have:

\sum_{i=1}^{26} a_i= 1(\frac{1-(1.06)^{26}}{1-1.06})= \frac{1-4.55}{-0.06}= 59.17.



Finally, multiplying 10,500 by 59.17 we have 621.285 ($).


The answer we found is very close to D. The difference can be explained by the accuracy of the values used in calculation, most important, in calculating (1.06)^{26}.


Answer: D



You might be interested in
Find the area of the region (in square units) exact number
Margaret [11]

The area of the region = area of the two rectangles = 64 units².

<h3>How to Find the Area of Rectangles?</h3>

A rectangle has a length and a width, and its area can be calculated using the formula below:

area of rectangle = (length)(width).

The region shown is composed of two rectangles. To find the area of the region, calculate the area of each of the rectangles in the image given.

Area of rectangle 1 = (length)(width)

Length = 12 units

Width = 4 units

Area of rectangle 1 = (12)(4)

Area of rectangle 1 = 48 units²

Area of rectangle 2 = (length)(width)

Length = 4 units

Width = 4 units

Area of rectangle 2 = (4)(4)

Area of rectangle 2 = 16 units²

The area of the region = area of the two rectangles = 48 + 16 = 64 units²

Learn more about the area of rectangle on:

brainly.com/question/25292087

#SPJ1

3 0
1 year ago
Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
zvonat [6]

Answer:

The maximum volume of a box inscribed in a sphere of radius r is a cube with volume \frac{8r^3}{3\sqrt{3}}.

Step-by-step explanation:

This is an optimization problem; that means that given the constraints on the problem, the answer must be found without assuming any shape of the box. That feat is made through the power of derivatives, in which all possible shapes are analyzed in its equation and the biggest -or smallest, given the case- answer is obtained. Now, 'common sense' tells us that the shape that can contain more volume is a symmetrical one, that is, a cube. In this case common sense is correct, and the assumption can save lots of calculations, however, mathematics has also shown us that sometimes 'common sense' fails us and the answer can be quite unintuitive. Therefore, it is best not to assume any shape, and that's how it will be solved here.

The first step of solving a mathematics problem (after understanding the problem, of course) is to write down the known information and variables, and make a picture if possible.

The equation of a sphere of radius r is x^2 + y^2 + z^2=r^2. Where x, y and z are the distances from the center of the sphere to any of its points in the border. Notice that this is the three-dimensional version of Pythagoras' theorem, and it means that a sphere is the collection of coordinates in which the equation holds for a given radius, and that you can treat this spherical problem in cartesian coordinates.

A box that touches its corners with the sphere with arbitrary side lenghts is drawn, and the distances from the center of the sphere -which is also the center of the box- to each cartesian axis are named x, y and z; then, the complete sides of the box are measured  2x,  2y and 2z. The volume V of any rectangular box is given by the product of its sides, that is, V=2x\cdot 2y\cdot 2z=8xyz.

Those are the two equations that bound the problem. The idea is to optimize V in terms of r, therefore the radius of the sphere must be introduced into the equation of the volumen of the box so that both variables are correlated. From the equation of the sphere one of the variables is isolated: z^2=r^2-x^2 - y^2\quad \Rightarrow z= \sqrt{r^2-x^2 - y^2}, so it can be replaced into the other: V=8xy\sqrt{r^2-x^2 - y^2}.

But there are still two coordinate variables that are not fixed and cannot be replaced or assumed. This is the point in which optimization kicks in through derivatives. In this case, we have a cube in which every cartesian coordinate is independent from each other, so a partial derivative is applied to each coordinate independently, and then the answer from both coordiantes is merged into a single equation and it will hopefully solve the problem.

The x coordinate is treated first: \frac{\partial V}{\partial x} =\frac{\partial 8xy\sqrt{r^2-x^2 - y^2}}{\partial x}, in a partial derivative the other variable(s) is(are) treated as constant(s), therefore the product rule is applied: \frac{\partial V}{\partial x} = 8y\sqrt{r^2-x^2 - y^2}  + 8xy \frac{(r^2-x^2 - y^2)^{-1/2}}{2} (-2x) (careful with the chain rule) and now the expression is reorganized so that a common denominator is found \frac{\partial V)}{\partial x} = \frac{8y(r^2-x^2 - y^2)}{\sqrt{r^2-x^2 - y^2}}  - \frac{8x^2y }{\sqrt{r^2-x^2 - y^2}} = \frac{8y(r^2-2x^2 - y^2)}{\sqrt{r^2-x^2 - y^2}}.

Since it cannot be simplified any further it is left like that and it is proceed to optimize the other variable, the coordinate y. The process is symmetrical due to the equivalence of both terms in the volume equation. Thus, \frac{\partial V}{\partial y} = \frac{8x(r^2-x^2 - 2y^2)}{\sqrt{r^2-x^2 - y^2}}.

The final step is to set both partial derivatives equal to zero, and that represents the value for x and y which sets the volume V to its maximum possible value.

\frac{\partial V}{\partial x} = \frac{8y(r^2-2x^2 - y^2)}{\sqrt{r^2-x^2 - y^2}} =0 \quad\Rightarrow r^2-2x^2 - y^2=0 so that the non-trivial answer is selected, then r^2=2x^2+ y^2. Similarly, from the other variable it is obtained that r^2=x^2+2 y^2. The last equation is multiplied by two and then it is substracted from the first, r^2=3 y^2\therefore y=\frac{r}{\sqrt{3}}. Similarly, x=\frac{r}{\sqrt{3}}.

Steps must be retraced to the volume equation V=8xy\sqrt{r^2-x^2 - y^2}=8\frac{r}{\sqrt{3}}\frac{r}{\sqrt{3}}\sqrt{r^2-\left(\frac{r}{\sqrt{3}}\right)^2 - \left(\frac{r}{\sqrt{3}}\right)^2}=8\frac{r^2}{3}\sqrt{r^2-\frac{r^2}{3} - \frac{r^2}{3}} =8\frac{r^2}{3}\sqrt{\frac{r^2}{3}}=8\frac{r^3}{3\sqrt{3}}.

6 0
3 years ago
A salesman's commission on 150,000 sale is 3000. what is the percent?
fomenos
The percentage is 98%.

You can find this by multiplying 98% (0.98) and 150,000 to get 147,000. Then subtract 147,000 from 150,000 to get 3,000.

I hope this helps!
8 0
3 years ago
How long does a person going 10 mph (16 kph ) to go .6 miles
Olegator [25]

the answer will be 5

8 0
3 years ago
The point ( 3,0) lies on a circle with the center at the origin. What is the area of the circle to the nearest hundreth
Lina20 [59]
1) Find the radius of the circle.  You can do that by inspection; no work required.
2) Use the "area of a circle" formula A = pi r^2 to calculate the area.
3) Round off your area calculation to the nearest hundredth.
3 0
3 years ago
Other questions:
  • Give an example of a linear equation in one variable with no solution. Explain how you know it has no solution.
    7·2 answers
  • What is the value represented by the letter A on the box plot of the data?
    14·1 answer
  • Sara got 98, 100, 65, 78, 98, 46, 100, 100, 45, and 50 on her reading test. Lee got 97, 67, 89, 99, 100, 45, 79, 89, 58, and 67
    5·1 answer
  • Como podria la logica contribuir con la convivencia social
    5·1 answer
  • Arrange the numbers from largest to smallest 2.6 to 2.061 2.61 20.66 6.21​
    5·2 answers
  • Solve 7+3(2g-6)= -29
    15·2 answers
  • Determine if the sequence is geometric. If it is, find the term named in the problem.
    9·1 answer
  • True or false?<br> please help me out
    12·2 answers
  • Am I'm correct I linked a picture with it​
    5·2 answers
  • HELP ME ⟟ WILL BRAINIEST
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!