Mathematics College

Simplify the expression 16 − x2 as much as possible after substituting 4 sin θ for x. (assume 0° < θ < 90°.)

Answers

Answer 1

The value of the expression [tex]16 - x^{2}[/tex] is 16 cos^2θ

Here,

The expression is [tex]16 - x^{2}[/tex]

We have to find the value of expression after substitute 4 sinθ for x.

What is substitution method?

Substitution method is algebraic method to solve the linear equations.

Now,

The expression is [tex]16 - x^{2}[/tex]

By putting x = 4 sinθ in above equation, we get;

[tex]16 - x^{2}[/tex] = 16 - ( 4 sinθ)^2

= 16 - 16 sin^2θ

= 16 ( 1 - sin^2θ )

= 16 cos^2θ

Since, 1 - sin^2θ = cos^2θ

Hence, The value of the expression [tex]16 - x^{2}[/tex] is 16 cos^2θ

Learn more about the substitution method visit:

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Answer 2

Final answer:

To simplify the expression 16 - x² after substituting 4 sin θ for x, you end up with 16 cos² θ, using the Pythagorean trigonometric identity.

Explanation:

To simplify the expression 16 - x² after substituting 4 sin θ for x, where 0° < θ < 90°, we follow these steps:

Substitute 4 sin θ into the expression for x.Simplify the resulting expression.

The substitution gives us 16 - (4 sin θ)².

To further simplify:

Calculate the square of 4 sin θ, which is 16 sin² θ.Subtract 16 sin² θ from 16, giving us 16 - 16 sin² θ.Factor out the common factor of 16, resulting in 16(1 - sin² θ).Recognize that (1 - sin² θ) is equal to cos² θ due to the Pythagorean identity.Finally, the simplified expression is 16 cos² θ.

Related Questions

which is the value that will make -7x<102

Answers

-7x < 102

First step is to divide both sides by -7. Because we are dividing by a negative number, we have to flip the signs.

-7x < 102
x > -102/7

The only one of your answer choices that is greater than -102/7 is A. 20.

Hope this helped!

The correct answer us 20. Option A.

how do you write 12.325 in expanded form

Answers

10 + 2 + 0.3 + 0.02 + 0.005

hope this helps

10.000+2.000+300+20+5

Which two fixed points define the shape of a hyperbola?

A.) center and focus
B.) focus and directrix
C.) two foci
D.) vertex and focus

Answers

C). The two foci points determine the shape of the hyperbola.

C is the answer learned this last year

one slice of cheese pizza contains 310 calories.A medium-size orange has one-fifth that number of calories. How many calories are in a medium-size orange?

Answers

310 dividedby 5. You do the math

slice of pizza has 310 calories
The orange has one-fifth of the pizza
So calories in orange = 310 ÷ 5 = 62 calories.

Your answer is 62 calories.

Hope it helped you. Mark as best!

60,488 rounded to the nearest 10,000

Answers

It would be 60,000 because

With the 488 it would be rounded for the nearest hundredth to 500 then if you had to round it to the nearest 1,000 from 60,500 it would be 61,000

So even with 61,000 rounded to the nearest 10,000 it would still be 60,000 in order to get 70,000 it would need to be 65,000 and anything above to be rounded to the 70,000 mark.

Last week, Donnell practiced the piano 3 hours longer than Marcus. Together, Marcus and Donnelly practiced the piano for 11 hours. For how many hours did each young man practice the piano?

Answers

Marcus practiced for 4 hours and Donnelley played 7 hours

Write the ratio 28 inches to 9 feet as a fraction in simplest form

Answers

28/108

In simplest form it would be: 7/27

least common multiple of 17,3,7

Answers

Least common multiple is 357.

the perimeter of a school crossing sign is 102 inches what is the length of each side

Answers

Add up all sides lengths:

s+6 + s+6 + s + s + 2s = 102

solve for s:

6s + 12 = 102
6s = 90
s= 15

The lengths of sides are as follows:

15, 21, 21, 15, 30 inches

I hope that helps!

Answer:

The length of each side is 21 inches, 21 inches, 15 inches, 30 inches and 15 inches.

Step-by-step explanation:

Consider the provided figure.

It is given that the perimeter of a school crossing sign is 102 inches.

The figure have 5 sides having the length s+6, s+6, s, 2s and s.

Now the perimeter is the sum of all sides,

s+6+s+6+s+2s+s=102

6s+12=102

6s=102-12

6s=90

s=15 inches

The length of each side is:

s+6=15+6=21 inches

s=15 inches

2s=2×15=30 inches

s=15 inches

Hence, the length of each side is 21 inches, 21 inches, 15 inches, 30 inches and 15 inches.

If an experiment consists of five outcomes with p(o1) = 0.10, p(o2) = 0.20, p(o3) = 0.30, p(o4) = 0.15, then p(o5) is

Answers

Given an experiment with 5 trials, the sum of probabilities is 1.

i.e.
p(o1) + p(o2) + p(o3) + p(o4) + p(o5) = 1

0.10 + 0.20 + 0.30 + 0.15 + p(o5) = 1

0.75 + p(o5) = 1

p(o5) = 1 - 0.75 = 0.25

p(o5) = 0.25

Are you shipping box has a height equal to its with which is equal to its length of the box can hold 1525 cm³ what is the length of one of its sides

Answers

check the picture below.

maggie’s brother is 6 years younger than four times her acne. the sum of their ages is 34. how old is maggie

Answers

m + b = 34
b = 4m - 6

m + 4m - 6 = 34
5m - 6 = 34
5m = 34 + 6
5m = 40
m = 40/5
m = 8 <=== Maggie's age

b = 4m - 6
b = 4(8) - 6
b = 32 - 6
b = 26 .....brothers age

Maggie is 13 because if you

What is 0.0004694 in scientific notation

Answers

It is 4.694 times 10^-4

This would be 4.694 * 10-4

Find an expression in simplest form for the verbal description.

The sum of the cost of dinner d and a $12 tip divided equally between 4 people

Answers

D+$3 that’s all you need

What property justifies 4-37-25=4-25-37

Answers

the commutative property

If a computer cost$600.00 and the sales tax rate is 7.5 percent, what is the total cost

Answers

7.5% of 600 = 45, total cost = 600+45 = $645

3 4/5 as a improper fraction may u pls explain how u get the answer

Answers

[tex]\bf \stackrel{\textit{mixed fraction}}{a\frac{b}{c}}\qquad \stackrel{\textit{improper fraction}}{\cfrac{a\cdot c+b}{c}} \\\\\\ thus\implies \stackrel{mixed}{3\frac{4}{5}}\implies \cfrac{3\cdot 5+4}{5}\implies \stackrel{improper}{\cfrac{19}{5}}[/tex]

A recipe calls for 400 grams of flour. if leena makes one quarter of the recipe, how many kilograms of flour will she need?

Answers

300 more grams of flour

Solve the equation:

(x^4 - x + y)dx - xdy=0

Answers

keeping in mind that "dx" is the delta x, or differential over the x-axis, whilst "dy" is the delta y or differential over the y-axis,

[tex]\bf (x^4-x+y)dx-xdy=0\implies \stackrel{\textit{distributing dx}}{x^4dx-xdx+ydx}-xdy=0 \\\\\\ ydx=xdy-x^4dx+xdx\implies y=\cfrac{xdy-x^4dx+xdx}{dx} \\\\\\ y=\stackrel{\textit{distributing the denominator}}{x\cfrac{dy}{dx}-x^4+x}[/tex]

Joelle is working on a school report in a word processor. She wants to place a figure which is 13.09 cm wide so that it appears centered on the page. The width of the page is 21.59 cm.

How much margin should she leave on the left so that the figure is centered?

Do I divide the width by 13.09 to get the answer? Not very sure

Answers

Alright, so for it to be centered, it needs to be x cm away from the left side and therefore x away from the right side (due to that if it's centered, it needs to be the same distance away from both), making the total width x+13.09+x=21.59=13.09+2x. Subtracting 13.09 from both sides, we get 8.5=2x and by dividing both sides by 2 we get x=3.25

Answer:

The answer is 4.25 cm.

Step-by-step explanation:

Given is - Joelle wants to place a figure which is 13.09 cm wide so that it appears centered on the page. The width of the page is 21.59 cm.

To solve, this first we will 13.09 from the 21.59, and then divide that number in half to find the left and right margin.

[tex]21.59-13.09=8.5[/tex]

[tex]\frac{8.5}{2}= 4.25[/tex]

Therefore, the answer is 4.25 cm.

Write the equation of the sphere in standard form. 2x2 + 2y2 + 2z2 = 12x − 24z + 1

Answers

The standard equation of a sphere with center C(a, b, c) and radius r is:

[tex](x-a)^2 + (x-b)^2 + (x-c)^2 = r^2[/tex]
Now, the given equation is :

[tex]2x^2 + 2y^2 + 2z^2 = 12x - 24z + 1 \\ 2x^2 + 2y^2 + 2z^2 - 12x + 24z - 1 = 0 \\[/tex]

combine the x, y and z terms and complete the square for each variable
[tex](2x^2 - 12x) + 2y^2 + (2z^2 + 16z) = 1 \\ (x^2 - 6x) + y^2 + (z^2 + 8z) = 1/2 [/tex]

complete square for x by adding 9 to both sides

we do not complete square for y because there is not a translation on that axis

complete square for z by add 16 to both sides
[tex] (x^2 - 6x + 9) + y^2 + (z^2 - 8x + 16) = 1/2 + 9 + 16 \\ (x - 3)^2 + y^2 + (z - 4)^2 = 25 1/2[/tex]

The standard equation of a sphere with center C(a, b, c) and radius r is:

[tex] (x-a)^{2}+ (y-b)^{2}+ (z-c)^{2}=r^{2}[/tex]

We have the equation:

[tex]2x^2+2y^2+2z^2=12x-24z+1[/tex]

which can be regrouped to fit the standard form, using completing the square as follows:

first divide both sides by 2

[tex]x^2+y^2+z^2=6x-12z+ \frac{1}{2} \\\\(x^2-6x)+y^2+(z^2+12z)= \frac{1}{2}\\\\(x^2-2\cdot3\cdot x)+y^2+(z^2+2 \cdot6 \cdot z)= \frac{1}{2}\\\\(x^2-2\cdot3\cdot x+3^2)-3^2+y^2+(z^2+2 \cdot6 \cdot z+6^2)-6^2= \frac{1}{2}\\\\(x-3)^2+y^2+(x+6)^2=9+36+0.5\\\\(x-3)^2+y^2+(x+6)^2=45.5\\\\\(x-3)^2+y^2+(x+6)^2=( \sqrt{45.5} )^2[/tex]

Answer: [tex](x-3)^2+y^2+(x+6)^2=( \sqrt{45.5} )^2[/tex]

Expand and simplify:
(x+2)(x+3)

Answers

The answer is x² + 5x + 6. All you have to do is distribute. I hope this helps love! :)

Prove by PMI that
3^n >= 1+2^n

Answers

Hello,

[tex]If\ n=1\ then\ 3^1 \geq 1+2^1\ since\ 3 \geq 3\\ \textrm{This proves the base case.}\\ \textrm{Now assume this holds for some n = k.}\\ \textrm{We need to show this holds for n = k +1}\\ 3^k \geq 1+2^k\ is\ true.\\ 3^{k+1}=3*3^k \geq 3*(1+2^k)\\ 3^{k+1}=3+(1+2)*2^k=3+2^k+2^{k+1} \geq 3+2^{k+1}\\ 3^{k+1} \geq 1+2^{k+1}\ is\ true. [/tex]

Noel Devine bought a new dishwasher for $320. He paid 20 down and made 10 monthly payments of $34 what actual yearly interest rate did Noel pay?

Answers

IN finding the interest we need to use the following formula:
Interest= Principal x Rate x Time or I= PRT
Substitute values: I= [$20 + ($10 x $34)] -320 =$40
P= $320
R=?
T=10 months/year
I=PRT
Since R is a missing term, we will solve for R using this formula: R=I/PT
R= [$20 + ($10 x $34)-320] / ($320 x 10 months)
T=10 months÷12 months=0.83
R= ($40)/ $320 X 0.83
R= 40/ 265.6
R=15.06024096

First we need to find the amount financed, total payments and total interest

Amount financed
320-20=300

Total payments
34×10=340

Total interest=total payments-amount financed
Total interest=340-300=40

Now to find the yearly interest rate use the formula of
I=(2yc)÷(m×(n+1))
I ?
Y number of months in a year 12
C total interest 40
M amount financed 300
N number of payments 10
I=(2×12×40)÷(300×(10+1))
I=0.2909×100=29.09%

Hope it helps!

is the number /2500 rational or irrational? explain

Answers

2500 is rational, because it can be turned into a fraction. A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. While a irrational number can be written as a decimal, but not as a fraction, and has endless non-repeating digits to the right of the decimal point.
Ex of rational: The number 8 is a rational number because it can be written as the fraction 8/1.
Ex of irrational: π = 3.141592…
Hoped it helped! :)

Answer:

Rational

Step-by-step explanation:

If you ask about number "2500" then it is rational since 2500 can be changed in the form of p/q, where q ≠ 0. Like, 2500, 5000/2 or so on.

and if you ask about "[tex]\frac{number}{2500}[/tex]" then it is also rational and it is terminating rational number because denominator (2500) can be converted in [tex]2^{m} *5^{n}[/tex].

a circle is centered at the point -3,2) and passes through the point (1,5). The radius of the circle is blank units. The point (-7, blank) lies on this circle

Answers

It's important that you share the complete question. What is your goal here? Double check to ensure that you have copied the entire problem correctly.

The general equation of a circle is x^2 + y^2 = r^2. Here we know that the circle passes thru two points: (-3,2) and (1,5). Given that a third point on the circle is (-7, ? ), find the y-coordinate of this third point.

Subst. the known values (of the first point) into this equation: (-3)^2 + (2)^2 = r^2. Then 9 + 4 = 13 = r^2.

Let's check this. Assuming that the equation of this specific circle is
x^2 + y^2 = r^2 = 13, the point (1,5) must satisfy it.

(1)^2 + (5)^2 = 13 is not true, unfortunately.

(1)^2 + (5)^2 = 1 + 25 = 26 (very different from 13).

Check the original problem. If it's different from that which you have shared, share the correct version and come back here for further help.

how many sets of 4 consecutive numbers are there such that the product of the numbers is less than 100000

Answers

The answer is 4x(12)<100000

In the number 555.55 each digit is a 5 but the value of each 5 is different based on its place in the number. How does the value of the 5 in the tenths place compare to the value of the underlined digit 5

Answers

Final answer:

The value of each 5 in the number 555.55 depends on its place value. A 5 in the tenths place (representing 1/10 or 0.1) would have a much smaller value compared to a 5 in the hundreds place in the same number, due to the multiplying factor of ten for each place value.

Explanation:

In the number 555.55, each digit is indeed a 5. However, the value of each 5 changes depending on its place value. The tenths place, to the right of the decimal point, represents tenths or 1/10 (0.1).

If the underlined digit 5 is in the hundreds place, for example, it will have a value of 500. To compare, the value of the 5 in the tenths place (which is 0.5) is much smaller than the value of the 5 in the hundreds place. In fact, it's 1,000 times smaller because each move to the left in place value multiplies the digit by 10.

Learn more about Place Value here:

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What power can you write to represent the volume of a cube shown? write the power as an expression with a base and an exponent and ,then find the volume of the cube.

Answers

a volume can be a great thing to use and a volume cube

By what percent will a fraction change if its numerator is decreased by 50% and its denominator is decreased by 25%?

Answers

(N/2)/(3D/4) = 2N/3D
So it is decreased by 1/3.

(N/2)÷(3D/4) = 2N/3D

So the answer is 33.33333% or 33 1/3%

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