A radioactive element decays by 40% every year. A scientist started an experiment with 5 grams of the radioactive material. Write a function that models the amount of radioactive material remaining after x years.
The amount of radioactive material that would remain by the 7th hour of decay is 0.12 grams.
What is Exponential function?The exponential function is a mathematical function denoted by f(x)=\exp or e^{x}. Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras.
here, we have,
An exponential function is in the form:
y = abˣ
where y,x are variables, a is the initial value and b is the multiplicative factor.
Let y represent the amount of substance remaining after x hours. Hence:
a = 9375, at x = 4, y = 15:
15 = 9375(b)⁴
b = 0.2
y = 9375(0.2)ˣ
For the 7th hour, x = 7:
y = 9375(0.2)⁷
= 0.12
The amount of radioactive material that would remain by the 7th hour of decay is 0.12 grams.
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Suppose an object moves along the y axis so that its location is yequalsf(x)equalsxsquaredplusx at time x (y is in meters, x is in seconds). Find (A) The average velocity (the average rate of change of y with respect to x) for x changing from 2 to 5. (B) The average velocity for x changing from 2 to 2+h. (C) The instantaneous velocity at xequals2 second. (A) The average velocity is 8 m/sec. (B) The average velocity is (nothing )m/sec. (C) The instantaneous velocity is 5 m/sec.
When x changes from 2 to 5, the average velocity (or average rate of change of y with respect to x) is 9.33 m/s. When x changes from 2 to 2 + h, the average velocity is 2 + hm/s.
How can the average velocity be determined?Average speed is determined by dividing the total distance you travelled by the total time, whereas average velocity is determined by dividing your displacement (a vector Throughout the entire period, position to your finish position).
(A) The following formula is used to get the average velocity when x changes from 2 to 5:
value of y at x = 2
= 2²+2 = 6 m
value of y at x = 5
= 5²+5 = 30 m
displacement equals 30 – 6 = 24 m.
Time is 5 - 2 = 3 seconds.
average velocity
= displacement /Time duration
= 24/3 = m/s
B) Value of y at x = 2 = 5 for x changing from x to x plus ie for minimal change in x.
Displacement during a little change in x will be 0 because the value of y will once again be 5.
C )
y(x) = x² + x
dy / dx = 2x + 1
dy / dx represents instantaneous velocity
so when x = 2
instantaneous velocity = 2 x 2 +1
= 5 m /s
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3) Find the sum of the first 50 terms. *
8 + 20+32 +44 + ...
15100
Step-by-step explanation :
Sum of first 50 terms for the AP:
[tex]\dashrightarrow { \boxed{\sf S = \dfrac{n}{2} (2a + (n - 1)d )}}[/tex]
where ,
Total no. of terms (n)= 50 Comman difference (d) = 20 - 8 = 12 First term (a) = 8On substituting the values , we get :
[tex]\dashrightarrow \sf S = \dfrac{50}{2}(2 \times 8 + (50 - 1) \times 12) \ \\ \\ \dashrightarrow \sf S = 25(16 + 49 \times 12 ) \\ \\ \dashrightarrow \sf S = 25(16 + 588 )\\ \\ \dashrightarrow \sf S = 25(604 )\\ \\ \dashrightarrow \sf S = 15100 \\ [/tex]
[tex] \therefore[/tex] Sum of the first 50 terms is 15100
One number is 7 more than another number. The sum of these two numbers is 101. What are the numbers?
Answer:
54 and 47
Step-by-step explanation:
Lets first number be "x" and second number be "y"
x = y + 7
x + y = 101
(y + 7) + y = 101
2y + 7 = 101
2y = 94
y = 47
x = 47 + 7
x = 54
Taking the earth to be the sphere of radius 6378000m, find the circumference of the equator of the earth. Give your answer in km and in terms of pi.
Answer:
12756pi km
Step-by-step explanation:
Circumference = pi * diameter
Diameter = radius * 2
kilometer = meter/1000
6378000*2= 12756000
diamter = 12756000 meters
12756000/1000 = 12756 kilometers
diameter = 12756 kilometers
12756pi km= circumference
is the expression 3(x+2) equivalent to 3 x+6? explain
Samuel took a survey of people entering several random ice cream shops to find out who liked vanilla and who liked chocolate ice cream.
biased or unbiased
Answer:
ummmm i would say unbiased fam only bc i love ice cream and i always wonder whichwtch one does people like more.
Step-by-step explanation:
The rate in the portion formula is not always expressed with a percent symbol?
That's correct. The rate in the portion formula can be expressed as a decimal, fraction, or percentage, depending on the context and what is most convenient for the problem being solved.
What is the proportional and ratio formula?How do you calculate ratio and proportion? Ans: The ratio formula is written as a: b a/b for any two numbers. Contrarily, the percentage formula is written as a:b::c:da:b::c:da:b=cd.
What does a ratio example look like?An equation in which two ratios are made equal is known as a percentage. As an illustration, you could express the ratio as 1: 3 if there is 1 guy and 3 girls (for every one boy there are 3 girls) 1 out of 4 are boys, and 3/ 4 are girls.
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At the local grocery store, A 12 ounce bottle of stay clean shampoo costs 8 and an 8 ounce bottle of shampoo costs 6. Which size bottle has the lesser cost per ounce?
Using their unit rates, the 12-ounce bottle costs lesser per ounce than the 8-ounce bottle of stay clean shampoo.
What is the unit rate?The unit rate is the cost per unit.
The unit rate is computed as the quotient of the total cost and the number of units.
The cost of a 12-ounce bottle of stay clean shampoo = $8
The unit rate per ounce of the 12-ounce bottle = $0.67 ($8/12)
The cost of an 8-ounce bottle of stay clean shampoo = $6
The unit rate per ounce of the 8-ounce bottle = $0.75 ($6/8)
Thus, comparing the unit rate of the 12-ounce bottle with the 8-ounce bottle, we can conclude that it costs more per ounce to buy the 8-ounce bottle than the 12-ounce bottle.
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Answer Quick
Factor what the image says
Answer:
49u² - 154u + 121 factored is (7u - 11)²
Step-by-step explanation:
To factor the given quadratic expression, 49u² - 154u + 121, rewrite in the form a² - 2ab + b².
Rewrite 49 as 7² and 121 as 11²:
[tex]\implies 7^2u^2-154u+11^2[/tex]
[tex]\textsf{Apply the exponent rule} \quad a^n \cdot c^n=(ac)^n:[/tex]
[tex]\implies (7u)^2-154u+11^2[/tex]
Rewrite 154u as 2 · 7u · 11 :
[tex]\implies (7u)^2-2 \cdot 7u \cdot 11+11^2[/tex]
Compare with the form a² - 2ab + b²:
a = 7ub = 11Apply the perfect square formula: a² - 2ab + b² = (a - b)²
[tex]\implies (7u)^2-2 \cdot 7u \cdot 11+11^2=(7u-11)^2[/tex]
Joel opened a
savings account and
deposited $3500 as
principal. The account
earns 7% interest,
compounded
semianually. What is the
balance after 48
months?
Answer:
Step-by-step explanation:
The balance of the savings account after 48 months can be calculated using the formula for compound interest:
A = P * (1 + r/n)^(nt)
where:
A is the balance after t years
P is the principal (initial amount)
r is the interest rate (as a decimal)
n is the number of times compounded per year
t is the number of years
Step-by-step, the calculation is as follows:
Convert the interest rate from a percentage to a decimal:
r = 7% = 0.07
Determine the number of times compounded per year:
n = 2 (compounded semi-annually)
Determine the number of years:
t = 48 months / 12 months/year = 4 years
Plug the values into the formula:
A = P * (1 + r/n)^(nt)
A = $3500 * (1 + 0.07/2)^(2 * 4)
Calculate the balance using the formula:
A = $3500 * (1.035)^8
A = $3500 * 1.306769
A = $4588.72
So, the balance in the savings account after 48 months is $4588.72.
write an equation for the function graphed.
The equation for the graphed quadratic function is:
y = -1*(x - 3)^2 + 2
How to write an equation for the function graphed?Here we have the graph of a quadratic equation. Remember that if the leading coefficient is a and the vertex is (h, k) then we can write:
y = a*(x - h)^2 + k
Here the vertex is at (3, 2) then we can write:
y = a*(x - 3)^2 + 2
We can see that it also passes through the point (1, -2), replacing these values we will get:
-2 = a*(1 - 3)^2 + 2
-2 = a*4 + 2
-2 - 2 = a*4
-4/4 = a = -1
Then the quadratic equation is:
y = -1*(x - 3)^2 + 2
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Metalworking Twelve equally spaced holes are to be drilled in a metal strip 3414-in. long, with 2 in. remaining on each end. What is the distance, to the nearest hundredth of an inch, from center to center of two consecutive holes.
Rounded to the nearest hundredth of an inch, the distance is 310.91 inches.
How would you define distance?How far apart or how far away items or locations are from one another is measured numerically as distance. A physical quantity, it is typically stated in terms of meters, kilometers, miles, or feet. Given that it simply has magnitude and no direction, distance is a scalar quantity.
Usually, a formula is used to determine the distance between two points. One example is the distance formula in Euclidean geometry, which includes computing the square root of the sum of the squares of the coordinate differences between the two points.
Distance can be measured in a variety of contexts, including physics, where it is used to describe an object's displacement through time, and geography, where it is used to define the separation between two points on the surface of the Earth. It is also frequently used in daily life when people go for walks or runs or when they drive from one area to another to calculate the distance they travel.
There are no holes in the first and last 2 inches of the metal strip, leaving 3410 inches (3414 - 2 - 2) as the remaining length.
We can divide the total remaining length by the number of gaps between the holes to determine the distance between the centers of two adjacent holes. The distance between the 12 holes is determined by the fact that there are 11 gaps between them.
distance = (total length - length of 2 end pieces) / (number of gaps)
distance = (3410 in.) / 11
distance ≈ 310.91 in.
Rounded to the nearest hundredth of an inch, the distance is 310.91 inches.
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The cost for shipping a package that weighs x pounds from Boise
to Los Angeles is shown at the right. How much more does
Shipping Central charge than Globe Delivery?
Shipping Central charges $0.51 more than Globe Delivery for any given weight of the package.
How to calculate this?The cost for shipping a package that weighs x pounds from Boise to Los Angeles is shown below.
Company Cost ($)
Shipping Central 3x+3.50
Globe Delivery 2x + 2.99
To determine the amount more Shipping Central charges than Globe Delivery, you can subtract the cost function for Globe Delivery from the cost function for Shipping Central.
The difference between the two cost functions is:
3x + 3.50 - (2x + 2.99) = x + 0.51 = $0.51 + x
This means that, for any given weight of the package, Shipping Central charges $0.51 more than Globe Delivery.
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LEVEL 3
Record the letters
in your triangle from
left to right.
Example:
ABCDEFGHI
Enter the correct 9 letter sequence (no spaces) Use all capital letters *
Triangle is a polygon with 3 sides. The record of the letters in your triangle from left to right is EGFBIHDCA.
What is a triangle?A triangle is a polygon with 3 sides. Also, It has three vertices.
As the example is given to us, ABCDEFGHI is the record of the letters that are when seen from top to bottom, therefore, if we rotate the triangle such that the left side of the triangle is on the upper side, then the triangle we will get is given below.
Thus, the record of the letters in your triangle from left to right is EGFBIHDCA.
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PLEASE HELP PLEASEPLEASEPLEASE
The angle measure m<T is equivalent to 78 degrees
Similar trianglesSimilar triangles are triangles that have similar sides and angles.
From the given triangle
<A = <R
<B = <S
<C = <T
Since the sum of angle in a triangle is 180 degrees, hence;
<A + <B + <T = 180
24 + 78 + <T = 180
102 + <T = 180
<T = 180 - 102
<T = 78 degrees
Hence the measure of m<T from the triangle is 78 degrees
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You made a big batch of slime for all of your friends. You have 7.2 cups of slime and 36 containers. What is the unit rate? (How much is in each container?) Enter your answer as a decimal in the box. The units are cups per container. Do not put the units.
Using the unitary method, we found that the unit rate of slime is 0.2 cups per container.
What is meant by the unitary method?
The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, and then multiplying that value to determine the required value. Recognizing the units and values is crucial when using the unitary technique to a problem.
Always write the items that need to be computed on the right side and the things that are known on the left side to simplify things. The unitary approach must be applied whenever we need to determine the ratio of one quantity to another.
Given,
Number of cups of slime = 7.2
Number of containers = 36
Unit rate = Amount of slime in each container
Using the unitary method we can find the unit rate.
7.2 cups = 36 containers
1 container = 7.2/36 = 0.2 cups.
Therefore using the unitary method, we found that the unit rate of slime is 0.2 cups per container.
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2-step Word problemas
Answer:
2) 47 snow globes (51+8-12) 23) 130 snowballs (46+59+25) 4) 92 penguins (83+69-55)
2023 Tax Year (Filing in 2024)
You are 25yr old and single with no kids/dependents.
You live in Florida.
Your job pays $84,649/yr.
You have no other sources of incomes.
Your itemized deductions come out to $10,211.
Your credits come out to $9.
Your tax-deffered retirement contributions come out to $3,288.
Calculate the total Federal Income Taxes and FICA paid.
If you are 25yr old and single with no kids/dependents. The total Federal Income Taxes is $11,218 and FICA paid is $6,478.
How to find the total Federal Income Taxes and FICA paid?First, you need to determine your taxable income.
Gross income: $84,649
Itemized deductions: $10,211
Tax-deferred retirement contributions: $3,288
Taxable income: $71,150
Next, you can use the tax brackets and rates for the 2023 tax year to calculate your federal income tax liability.
The first $10,950 is taxed at 10%: $1,095
The next $32,250 is taxed at 12%: $3,870
The next $28,425 is taxed at 22%: $6,253
The remaining $0 is taxed at 24%: $0
Total federal income tax liability: $11,218
b. For 2023, the Social Security tax rate is 6.2% on the first $147,000 of income, and the Medicare tax rate is 1.45% on all income.
Social Security tax: $5,240.80
Medicare tax: $1,237.20
Total FICA taxes: $6,478
Therefore the total Federal Income Taxes is $11,218 and FICA paid is $6,478.
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What is the x-intercept of this equation? 3x + 5y = 9
To find the x-intercept of the equation 3x + 5y = 9, we set y = 0 and solve for x.When y = 0, the equation becomes 3x + 5(0) = 9, which simplifies to 3x = 9.Solving for x, we get x = 3.The x-intercept of the equation 3x + 5y = 9 is (3, 0).
What is x-intercept ?
An x-intercept is a point where a graph or a line crosses the x-axis. It is the value of the independent variable (x) when the dependent variable (y) is equal to zero. In other words, it is the point at which a function or equation intersects the x-axis, and its coordinates are of the form (x, 0). To find the x-intercept of a graph or equation, you can set y=0 and solve for x.
To find the x-intercept of the equation 3x + 5y = 9, we need to set y = 0 and solve for x.
So, we have:
3x + 5(0) = 9
3x = 9
x = 3
Therefore, the x-intercept of the equation 3x + 5y = 9 is (3,0), which means the point where the line intersects the x-axis is (3,0).
To verify, we can graph the equation and see that the point (3,0) is indeed on the x-axis.
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22.1.3 Quiz: Selecting a House: Fairly Priced?
Question 5 of 10
A house is listed for sale at $235,000, but the listing does not include square
footage of the house. Based on the comps, the line of best fit is
y=0.06x + 60.5. If the price is fair, what size (in square feet) should the house
be?
OA. 2850 ft2
OB. 2900 ft2
OC. 20,000 ft²
OD. 2350 ft²
Size of the house should be 2900 ft².
Correct option is B.
What is linear equation?A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation.
Given,
Equation for the line that expresses price of house
y = 0.06x + 60.5
where y is the price in thousand dollars, x is area in square feet.
Price of the house = $235000
⇒ y = 235
Then,
235 = 0.06x + 60.5
0.06x = $235 - 60.5
0.06x = 174.5
x = 174.5/0.06
x = 2908.33 ≈ 2900
Hence, 2900 ft² should be the size of the house.
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The measure of angle 4 is 120°, and the measure of angle 2 is 35°.
Triangle A B C. Angle A is 2, angle B is 1, angle C is 3. The exterior angle to angle B is 5, and the exterior angle to angle C is 4.
What is the measure of angle 5?
95°
105°
130°
155°
The measure of angle that is an exterior angle to B is 95°.
What are exterior angles?The angle created by any extended side of a triangle and its neighboring side is referred to as the external angle of a triangle. A triangle has three external angles. It should be noticed that every outside angle and corresponding interior angle constitute a linear pair. We are aware that a triangle's internal angle is created when its sides come together at its apex.
Given that,
The measure of angle 4 is 120°, and the measure of angle 2 is 35°.
Triangle A B C. Angle A is 2, angle B is 1, and angle C is 3. The exterior angle to angle B is 5, and the exterior angle to angle C is 4.
The sum of an interior and exterior angle is 180°,
∠4 + ∠3 = ∠1 + ∠5 = ∠2 + ∠6 = 180°
and ∠2+ ∠1 +∠3 = 180°
Also, ∠3 = 180 - ∠4
∠3 = 180 - 120 = 60°
Substitute the values,
∠2+ ∠1 +∠3 = 180°
35 + ∠1 + 60 = 180
∠1 = 180 - 95
∠1 = 85°
Now, ∠1 + ∠5 = 180
∠5 = 180 - 85
∠5 = 95°
Hence, the measure of angle that is an exterior angle to B is 95°.
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The correct question is:
Answer: A) 95 Degrees
Step-by-step explanation:
for each inequality write all the one digit integers that satisfy it a>2
Possible answers are:
3
4
5
6
7
8
9
Find the value of the variables. If the answer is not an integer, express it in simplest radical form.
60
20
Please help
The values of x and y would be [tex]x=\frac{20 \sqrt{3}}{3}[/tex] and [tex]y=\frac{40\sqrt{3}}{3}[/tex].
What are trigonometric ratios?
Trigonometric ratios are mathematical functions that describe the relationship between the angles and sides of a right triangle. The three main trigonometric ratios are the sine, cosine, and tangent.
In the given figure, by using trigonometric ratios, we can find x
tan60° = 20/x
√3 = 20/x
x = 20/√3
=> [tex]x=\frac{20 \sqrt{3}}{3}[/tex]
Now by using the ratio sin60°
sin60° = 20/y
√3/2 = 20/y
y = 40/√3
[tex]y=\frac{40\sqrt{3}}{3}[/tex]
Hence, the values of x and y would be [tex]x=\frac{20 \sqrt{3}}{3}[/tex] and [tex]y=\frac{40\sqrt{3}}{3}[/tex].
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Write the equation of the line in all three forms that is perpendicular to the line
6x+18y=36 and goes through the point (-5,4)
.
Find the amount of money (Future Value) in an account where $5,100 is deposited (Present Value) at an interest rate of 8% per year compounded continuously and the money is left in the account for 13 years.
round to 2 decimal places.
Answer:
Step-by-step explanation:
The formula for calculating the future value of a continuously compounded investment is:
FV = PV * e^(rt)
where PV is the present value, r is the annual interest rate as a decimal, t is the time in years, and e is the mathematical constant approximately equal to 2.718.
Plugging in the given values, we get:
FV = 5100 * e^(0.08 * 13) = 5100 * e^1.04 = 5100 * 2.856116786 = 14536.86
So, the future value of the investment after 13 years at an 8% interest rate compounded continuously would be approximately $14,536.86. Rounded to 2 decimal places, this is $14,536.86.
The average January temperature in your city is 45° F, but the actual temperature can vary up to 5° F warmer or colder depending on the year. Write an absolute value inequality that expresses the actual temperature in your city.
An absolute value inequality that expresses the actual temperature in your city, given that the average January temperature is 45° F and the temperature can vary up to 5° F warmer or colder, would be: | x - 45 | ≤ 5.
Where x represents the actual temperature in degrees Fahrenheit.
What is absolute value inequality?An inequality with an absolute value expression is referred to as an absolute value inequality. A quantity is enclosed in vertical bars (| |), which stand for the distance between that quantity and zero on the number line, to indicate an absolute value statement. The absolute value of an integer is always positive, hence regardless of the sign of x, |x| indicates the distance of x from zero.
Finding all feasible values of the variable that meet the inequality is the goal when resolving an absolute value inequality. To accomplish this, we first isolate the expression defining the absolute value, and then divide the inequality into two cases, one for positive and one for negative expressions defining the absolute value. Then, we resolve each case separately and combine the solutions.
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HELPPPP
Mr. Hagan writes an equation on the board.
8k−14=−4k+10+12(k−2)
Which statement is true about Mr. Hagan’s equation when solving for k?
A. The equation has exactly one solution.
B. The equation has exactly two solutions.
C. The equation has infinitely many solutions.
D. The equation has zero solutions.
C - this equation is simplified to a numeric identity. the variable has infinite solutions.
8k - 14 = -4k + 10 + 12(k - 2) --> use distributive property
8k - 14 = -4k + 10 + 12k -24 --> combine like terms
8k - 14 = 8k -14 --> combine like to each side
8k - 8k = -14 + 14 --> simplify
0 = 0
note, the variable cancels out, therefore there is no identity and ANY numeroc value can be used, as ant value will cancel itself out.
this means there is an infinite numbers of solutions.
Miguel Rodriguez borrowed $200 from his brother Julio to pay for books and tuition. He agreed to pay Julio in 6 months with simple annual interest at 6.3%
Answer:
simple interest is $6.8
Miguel pay the amount to Julio at the end of 6 months was $206.8
Step-by-step explanation:
Given : Miguel Rodriguez borrowed $200 from his brother Julio to pay for books and tuition. He agreed to pay Julio in 6 months with simple annual interest at 6.8%.
To Find : a) How much will the interest amount to?
b) What amount must Miguel pay Julio at the end of 6 months?
Solution :
Formula for simple interest :
Where rate is in decimal
Time is in years
So, rate of interest = 6.8 % = 0.068
Time = 6 months = 6/12=0.5
Principal = $200
Thus= 200x0.068x0.5=6.8
Q: What is the result of adding
-3x²-5x +1 and 8x²-2x-9?
A. 5x²-3x-10
B. -5x²-3x-10
C. 5x²-7x-8
D. -5x²-7x+8
Answer:
C. 5x²-7x-8
Step-by-step explanation:
(-3x²-5x +1 ) + (8x²-2x-9)
Grouping like terms:
(- 3x² + 8x²) + (- 5x - 2x) + (1 - 9)
=> (5x²) + (-7x) + (-8)
=> 5x² - 7x - 8
