Samuel paid $720 for a new music system. The list price for the music system was $800. What was his rate of discount?

Answer: [tex]5\sqrt{2}\ in[/tex]

Step-by-step explanation:

Given: In a 45-45-90 triangle, the length of one of the legs= 5 in.

Let H be the hypotenuse of the given right triangle.

Since the two angles are equal (45°) in the given right triangle,also in a triangle the side opposite to the equal angles are equal.

Therefore, the other leg of given right triangle = 5 in.

Now applying Pythagoras theorem, we get

[tex]H^2=5^2+5^2=25+25\\\\\Rightarrow\ H^2=50\\\\\Rightarrow H=\sqrt{50}\\\\\Rightarrow H=5\sqrt{2}\ in[/tex]

Hence, the length of the hypotenuse = [tex]5\sqrt{2}\ in[/tex]

Answer:

greatest common factor the largest factor that any given numbers have in common

Step-by-step explanation:

so the answer is greatest common factor

Final answer:

Knowledge of prime and common factors is essential for finding the greatest common divisor (GCD) or greatest common factor (GCF) of two or more numbers, a key element in simplifying fractions and understanding ratios.

Explanation:

Understanding prime and common factors is instrumental in finding the greatest common divisor (GCD) or greatest common factor (GCF) of two or more numbers. The GCD is the highest number that divides evenly into each of the numbers in question. For example, the GCD of 48 and 60 is 12 since 12 is the largest number that can divide both 48 and 60 without leaving a remainder. Finding the GCD is crucial in simplifying fractions, solving problems involving ratios, and is also a fundamental concept in more advanced mathematics including algebra and number theory.

The process often involves factoring the numbers down to their prime factors, identifying the common primes, and then multiplying these common prime factors together to find the GCD. This method not only highlights the importance of prime numbers in the mathematical hierarchy but also underlines the indelible role of factorization in simplifying and solving numerical problems.

Answer:

The number of steps in a stairway is discrete.

Step-by-step explanation:

A discrete data is that, which can take only integer values like number of apples in a basket.

But the continuous data is that, which can take on any value like a decimal. For example, the weight of patients in a hospital.

So, the number of steps in a stairway will be a discrete data. As stairs are counted in whole numbers and not decimals.

Solution for Q1:

Given is the height of ice-cream cone, h = 6 inches.

Given is the diameter of ice-cream cone, d = 2 inches. Then radius of cone would be half of diameter, r = d/2 = 1 inch.

The ice-cream without scoop would be in the shape of cone only. So we need to find the volume of cone.

We know the formula for volume of cone is given as follows :-

[tex] Volume = \frac{1}{3} \pi r^{2} h \\\\Volume = \frac{1}{3} \pi (1)^{2} (6) \\\\Volume = \frac{6\pi}{3} = 2\pi = 2(3.14) = 6.28 \;cubic \;inches. [/tex]

So the volume is 6.28 ≈ 6 in³.

Hence, option A is correct i.e. 6 cubic inches.

Solution for Q2:

A candy is in shape of sphere with diameter, d = 0.25 inches.

Then radius of sphere, r = d/2 = 0.250/2 = 0.125 inches.

The formula for volume of sphere is given as follows :-

[tex] Volume = \frac{4}{3} \pi r^{3} \\\\Volume = \frac{4}{3} \pi (0.125)^{3} \\\\Volume = 0.00818123 \;cubic \;inches. [/tex]

So, volume of one candy = 0.00818123 cubic inches.

There are total 120 candies in the machine.

Total volume of all 120 candies = 120 x 0.00818123 = 0.981748 cubic inches.

Hence, the total volume of all candies in the machine is about 0.982 in³.

Answer:

This should definitely help for question 1 :)

What is the volume of a ball if its diameter is 6 inches? Leave your answer in terms of π.

A)V = 12 π cubic inches

B)V = 36 π cubic inches

C)V = 288 π cubic inches

D)V = 27.75 π cubic inches

Explanation:The volume of the ball above is V = 36 π cubic inches. Use the formula V = 4/3π 3. Just remember to cut the diameter in half to get the radius.

Step-by-step explanation:

Answer:

360°

Step-by-step explanation:

I just took the test.

Answer:

[tex]^{10}C_5(2a)^{5}(-3b)^5[/tex]

Option 1 is correct.

Step-by-step explanation:

Given: [tex](2a-3b)^{10}[/tex]

We need to find 6th term of binomial expansion.

Formula:

[tex]T_{r+1}=^nC_rx^{n-r}y^{r}[/tex]

We need to find 6th term, [tex]T_6[/tex]

[tex]T_6=T_{r+1}[/tex]

So, r=5

Put r=5 into formula

[tex]n=10, x=2a \text{ and }y=-3b[/tex]

[tex]T_6=^{10}C_5(2a)^{10-5}(-3b)^5[/tex]

Now we will simplify it to calculate 6th term

[tex]T_6=^{10}C_5(2a)^{5}(-3b)^5[/tex]

Hence, The 6th term of binomial is [tex]^{10}C_5(2a)^{5}(-3b)^5[/tex]

The expression that represents the sixth term in the binomial expansion of (2a - 3b)¹⁰ is expressed as  ¹⁰C₆(2a)⁴(-3b)⁶.

To find the sixth term in the binomial expansion of (2a - 3b)¹⁰, we can use the binomial theorem,

which states that the k-th term in the expansion of (a + b)ⁿ can be calculated using the following formula:

T(x) = C(n, x) × (aⁿ⁻ˣ) × (bˣ)

Where:

T(x) is the x-th term in the expansion.

C(n, x) is the binomial coefficient, also denoted as "n choose x," and it is equal to n! / (x! × (n-x)!), where "!" denotes factorial.

a is the first term in the binomial (2a in this case).

b is the second term in the binomial (-3b in this case).

n is the exponent of the binomial (10 in this case).

x is the term we want to find (the sixth term in this case).

Now, let's calculate the sixth term:

x = 6

n = 10

a = 2a

b = -3b

First, calculate the binomial coefficient:

C(10, 6) = 10! / (6! × (10-6)!)

C(10, 6) = 210

Now, plug these values into the formula:

T(6) = 210 × (2a)¹⁰⁻⁶ × (-3b)⁶

Simplify the exponents:

T(6) = 210 × (2a)⁴ × (-3b)⁶

T(6) = 210 × 16a⁴× 729b⁶

Now, multiply the constants:

T(6) = 30,240a⁴b⁶

Therefore, the sixth term in the binomial expansion of (2a - 3b)¹⁰ is 30,240a⁴b⁶ which is written as ¹⁰C₆(2a)⁴(-3b)⁶.

learn more about binomial expansion here

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Answer:

circle 1 is a dilation of circle 2 with a scale factor of 2.5.

Step-by-step explanation:

The radius of circle 1 is 5x and the radius of circle 2 is 2x.

This means the scale factor of the dilation is 5x/2x = 2.5.

Circle 1 is a dilation of circle 2 with a scale factor of 0.4.

Two figures are said to be similar if they have the same shape and the ratio of their corresponding sides are in the same proportion.

Given that circle 1 has radius of 5x and circle 2 has radius of 2x, hence:

Circle 1 scale factor = 2x / 5x = 0.4

Hence:

Circle 1 is a dilation of circle 2 with a scale factor of 0.4.

Find out more on similar figures at: https://brainly.com/question/14285697

The confirmed correct answer is -9%. Trust me.

Answer:

Option B = -9%

Step-by-step explanation:

Given : Given an exponential function for compounding interest, [tex]A(x) = P(.91)^x[/tex]

To find : What is the rate of change?  

Solution :

The general form of the exponential function is [tex]f(x)=a(1+r)^x[/tex]

Where,

r is the rate of change

if r> 1 then it is growth rate    

if r< 1 then it is decay rate.

Comparing given function with exponential function,

[tex]A(x) = P(.91)^x[/tex]

[tex]1+r=0.91[/tex]

[tex]r=0.91-1[/tex]

[tex]r=−0.09[/tex]

It is a decay rate.

Convert into percentage, multiply with 100

[tex]-0.09\times 100=-9\%[/tex]

Therefore, Option B is correct.

The rate of change is -9%.      

Answer:

c

Step-by-step explanation:

Answer:

Approximately 157.08 sq. cm.

Step-by-step explanation:

The Area of an Ellipse is given by the formula  [tex]A=\pi a b[/tex]

Here 2a = 10, so a = 5

Also, 2b = 20, b = 10

Substituting these into the area formula gives us:

[tex]A=\pi a b\\A=\pi (5)(10)\\A=50\pi\\A=157.08[/tex]

The Area is approximately 157.08 sq. cm.

Answer:

Option D is correct

Step-by-step explanation:

Given Equations of Circles:

Circle 1 - [tex](x+12)^2+(y+16)^2=(r_1)^2[/tex]

Circle 2 -  [tex](x-30)^2+(y-16)^2=(r_2)^2[/tex]

Both circles passes through origin.

To find: Values of [tex]r_1\:,\:r_2[/tex]

Coordinates of origin = ( 0 , 0 )

Circles passes through origin means x = 0 & y = 0 must satisfy the equation of circles.

So, Substituting x = 0 &  y = 0 in Eqn of Circle 1

we get

[tex](0+12)^2+(0+16)^2=(r_1)^2[/tex]

[tex](r_1)^2=12^2+16^2[/tex]

[tex](r_1)^2=144+256[/tex]

[tex](r_1)^2=400[/tex]

[tex]r_1=\sqrt{400}[/tex]

[tex]r_1=20[/tex]

Now, Substituting x = 0 &  y = 0 in Eqn of Circle 2

we get

[tex](0-30)^2+(0-16)^2=(r_2)^2[/tex]

[tex](r_2)^2=(-30)^2+(-16)^2[/tex]

[tex](r_2)^2=900+256[/tex]

[tex](r_2)^2=1156[/tex]

[tex]r_2=\sqrt{1156}[/tex]

[tex]r_2=34[/tex]

Therefore, Option D is correct .i.e., [tex]r_1=20\:\:,\:\:r_2=34[/tex]

Answer is -2x^2+8-10 hope this helps!!!!

Answer:

26

Step-by-step explanation:

3 x 8 = 24

50-24=26

She will get $26 dollars back

Easy Step by Step explanation

Answer:

The answer is 5.

Step-by-step explanation:

Answer:

[tex]\text{Hence, area of regular pentagon is }32.7 cm^2[/tex]

Step-by-step explanation:

Given that a regular pentagon has an apothem measuring 3 cm and a perimeter of 21.8 cm.

we have to find the area of pentagon.

Apothem=a=3 cm

Perimeter=s=21.8 cm

[tex]\text{Area of regular pentagon=}\frac{1}{2}\times apothem\times perimeter[/tex]

[tex]\frac{1}{2}\times s \times a[/tex]

[tex]=\frac{1}{2}\times 21.8\times 3[/tex]

[tex]=\frac{65.4}{2}=32.7 cm^2[/tex]

[tex]\text{Hence, area of regular pentagon is }32.7 cm^2[/tex]

Option 3 is correct.

Answer:

The rocket is at a greater height when t = 3 than it is when t = 1.

Option B

Step-by-step explanation:

Just took the test

Answer:

B

Step-by-step explanation:

!

Given is the borrowed amount = 380 dollars.

Repayment period = 6 months.

Monthly installments = 67 dollars.

After six months, total repaid amount would be sum of all six monthly installments i.e. 6 x 67 = 402 dollars.

The interest to be paid on this loan would be difference of total repaid amount and borrowed amount.

Interest paid = Repaid amount - Borrowed amount

Interest = 402 dollars - 380 dollars

Interest = 22 dollars.

So, he would pay 22 dollars as interest on this loan.

The length of the diagonal of a square with side lengths 7√2 is approximately 14.1 units.

The length of the diagonal of a square can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides. Since a square has all sides equal, the length of each side can be represented as 7√2. Substituting this value into the Pythagorean theorem, we get:

Diagonal² = (7√2)² + (7√2)²

Diagonal² = 98 + 98 = 196

Taking the square root of 196, we find the length of the diagonal to be approximately 14 units. Rounding to the nearest tenth gives us 14.1 units.