Answers
We need to find how much will be left after 6 half-lives of a radioactive isotope starting with 130g.
One way to write the amount N of radioactive isotope left after a time t, with an initial amount N₀ and a half-life τ is:
[tex]N=N_0\left(\frac{1}{2}\right)^{t\text{ /}\tau}[/tex]Notice that when t = τ, we have:
[tex]N=\frac{N_0}{2}[/tex]In this problem, we have:
[tex]\begin{gathered} N_0=130g \\ \\ t=6\tau \end{gathered}[/tex]Then, we obtain:
[tex]N=130g\left(\frac{1}{2}\right)^{6\tau\text{ /}\tau}=130g\left(\frac{1}{2}\right)^6=\frac{130g}{64}\cong2\text{ g}[/tex]Therefore, rounding to the nearest gram, the answer is 2 grams.
Related Questions
given a function's domain values: -9, -6, -3, 0, 1.5, and 3, what is the range of the function's inverse?
Answers
Answer:
[tex]-9,-6,-3,0,1.5,3[/tex]Explanation:
Given the domain of the function as;
[tex]-9,-6,-3,0,1.5,3[/tex]Note that the range of the inverse of a function is the same as the domain of a function is the original function.
So the range of the inverse of the function will be;
[tex]-9,-6,-3,0,1.5,3[/tex]6) 3x2 + 10X - 8We have to find the vertex
Answers
Given:
[tex]3x^2+10x-8[/tex]To find the vertex, use the vertex formula below:
[tex]y\text{ = }a(x-h)^2+k[/tex]Where the vertex is: (h, k)
Thus, we have:
[tex]y=3x^2+10x-8[/tex][tex]\begin{gathered} y+8=3x^2+10-8+8 \\ \\ y+8=3x^2+10 \end{gathered}[/tex]Factorize:
[tex]undefined[/tex]What is an equation of the line that passes through the points (4,7) and (2,-1)? Answer in fully reduced form.
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The eqaution of a line between two points is given by:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]Plugging our points we have:
[tex]\begin{gathered} y-(-7)=\frac{-1-(-7)}{2-4}(x-4) \\ y+7=\frac{6}{-2}(x-4) \\ y+7=-3(x-4) \\ y+7=-3x+12 \\ y=-3x+12-7 \\ y=-3x+5 \end{gathered}[/tex]Therefore the equation is:
[tex]y=-3x+5[/tex]During a 60 minute boxing class Abby burns 480 calories, on a 40 minute run, Abby burns 440 calories, while cycling for 30 minutes which exercise is the most efficient at burning calories?
Answers
Explanation:
Given that;
In a 60 minute boxing class Abby burns 480 calories.
The rate at which calories are burn during boxing is;
[tex]\begin{gathered} r_1=\frac{480}{60} \\ r_1=8\text{ calories/minute} \end{gathered}[/tex]Also;
on a 40 minute run,Abby burns 440 calories,
The rate at which calories are burn during cycling is;
[tex]\begin{gathered} r_2=\frac{440}{40} \\ r_2=11\text{calories/minute} \end{gathered}[/tex]while cycling for 30 minutes,
From the above rate the most efficient exercise at burning calories is the exercise with the highest rate of burning calories.
angie drew a rectangle. the length of a rectangles she drew is 2 less than three times the width. find the dimensions of the rectangle if the rectangle if the area is 65 square meters
Answers
Answer:
The width of the rectangle is 5
The length of the rectangle is 13
Explanation:
Let's call x the length of the rectangle and y the width of the rectangle.
The length is 2 less than 3 times the width, so
x = 3y - 2
And the area is 65 square meters. Since the area is length times width, we get:
xy = 65
Now, we can replace the first equation x = 3y - 2 on the second one to get
(3y - 2)y = 65
3y(y) - 2y = 65
3y² - 2y = 65
3y² - 2y - 65 = 0
So, using the quadratic equation, we get that the solutions to 3y² - 2y - 65 = 0 are
[tex]\begin{gathered} y=\frac{-(-2)\pm\sqrt[]{(-2)^2-4(3)(-65)}_{}}{2(3)} \\ y=\frac{2\pm\sqrt[]{784}}{6} \\ y=\frac{2\pm28}{6} \\ \text{Then} \\ y=\frac{2+28}{6}=\frac{30}{6}=5 \\ or \\ y=\frac{2-28}{6}=\frac{-26}{6}=-\frac{13}{3} \end{gathered}[/tex]The solution is y = 5 because the width can't have a negative length.
Then, replacing y = 5 on the first equation, we get:
x = 3y - 2
x = 3(5) - 2
x = 15 - 2
x = 13
Therefore, the length of the rectangle is 13 meters and the width of the rectangle is 5 meters
Which linear function has the greater rate of change, the one described by the equation y = 3x + 4 or the one described by the table? 1 WN X у 3 10 17 24 4 Choose the correct answer below. O A. The function for the table has the greater rate of change. OB. The function y = 3x + 4 has the greater rate of change.Table:x y1 32 103 174 24
Answers
The rate of change is equivalent to the slope in linear functions.
So, the bigger the slope, the greater the rate of change.
In the case of the linear function y=3x+4, the slope is m=3.
In the case of the table, we have to select two points in order to calculate the slope m.
We will pick the points (1,3) and (2,10).
Then, we can calculate the slope as:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{10-3}{2-1}=\frac{7}{1}=7[/tex]As the slope of the function in the table is bigger than the slope of the function y=3x+4, the function from the table has a greater rate of change.
[tex]\begin{gathered} m_t>m_f \\ 7>3 \end{gathered}[/tex]Answer: Option A (The function for the table has the greater rate of change)
rewrite using a single positive exponent (7^8)/(7^5)
Answers
Given
[tex]\frac{7^8}{7^5}[/tex]When you divide two exponents with the same base number, to simplify the expression you have to calculate the difference between the index from the numberator and the index from the denominator.
In this case the base number is "7"
The index of the numerator is "8"
The index of the denominator is "5"
You can simplify the expression as follows
[tex]\frac{7^8}{7^5}=7^{8-5}=7^3[/tex]The solution is
[tex]7^3[/tex]When information is presented in the form of a bar graph or time- series graph, you could get more exact values if all the data were just listed out in table form. Then why not always do that. Why bother with graphs?
Answers
Step 1:
Graphs are essentially a visual display of quantitative information along two axes. Visuals are used as a way for our brains to quickly understand information, which is a powerful tool if used correctly. Graphs can show a large amount of data quickly in a way that is easy to process, without distracting people with a bunch of numbers.
two cyclists, 112 miles apart riding toward each other at the same time. One cycles 3 times as fast as the other. if they meet 4 hours later, what is the speed (in mi/h) of the faster cyclist?
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distance= 112 miles apart
time = 4 hours
Speed of a = 3 *speed of b
Distance covered by a = 3 * distance covered by b
Speed = distance / time
3b+b = 112
4b = 112
b= 112/4
b= 28 miles
a = 3b
a = 3 *28
a= 84 miles
Since both travelled for 4 hours:
Speed of A = 84 /4 = 21 mph
Speed of b= 28 /4 = 7 mph
Speed of the faster cyclist : 21 mph
The perimeter of triangle XYZ is 24 units.
What is the area of triangle XYZ? Round to the nearest
tenth of a square unit.
Trigonometric area formula: Area=1/2absin(C)
o 14.7
square units
14.9 square units
15.0 square units
15.3 square units
Answers
To the nearest tenth, the area of triangle XYZ is approximately 14.7 units².
What is a triangle?A triangle is a 3-sided polygon that is occasionally (though not frequently) referred to as the trigon.
There are three sides and three angles in every triangle, some of which may be the same.
Triangles can be divided into three groups based on the lengths of their sides, and these groups are as follows: Scalene, Isosceles, and Equilateral.
So, the area of the triangle is:
We know that:
<YXZ = 102°
Length of XY = z = 3
Length of YZ = x = 11
Length of XZ = y = ?
The perimeter of the triangle:
x+y+z = 24 units
Length XZ = y:
11+y+3 = 24
14+y = 24
y = 24 - 14
Therefore, y = 0.
Area of triangle XYZ:
The formula for trigonometric area is area of XYZ = 12(yz)sin X.
Putting values of x, y, and z as follows:
Area of ∆ XYZ = ½×10×3×sin 102
= ½×30×0.9782
= 15 × 0.9782
= 14.673
Therefore, to the nearest tenth, the area of triangle XYZ is approximately 14.7 units².
Know more about triangles here:
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Correct question:
What is the area of triangle XYZ? Round to the nearest tenth of a square unit. Trigonometric area formula: Area 14.7 square units 14.9 square units 15.0 square units 15.3 square units
Question Sally, an investor, purchases 3,000 shares in company X at $1.75 per share. After purchasing the shares the share price increases to $2.25 per share, after which Sally decides to sell her shares. Sally is required to pay 25% tax on all profits that she makes from the sale of the shares (called Capital Gains tax). Calculate the amount of tax that Sally must pay. Give your answer to the nearest dollar. Give your answer in dollars without the dollar sign or commas.
Answers
We are given the following information
Number of shares = 3,000
Buying price of a share = $1.75
Selling price of a share = $2.25
Capital Gains tax = 25% = 0.25
We are asked to calculate the amount of tax that Sally must pay.
Let us first calculate the profit.
Profit is given by
Profit = Selling price - Buying price
The buying price is given by
Buying price = (Number of shares)×(Buying price of a share)
Buying price = 3,000×1.75
Buying price = $5,250
The selling price is given by
Selling price = (Number of shares)×(Selling price of a share)
Selling price = 3,000×2.25
Selling price = $6,750
Profit = Selling price - Buying price
Profit = $6,750 - $5,250
Profit = $1,500
Finally, the amount of tax is given by
Amount of tax = profit × Capital Gains tax
Amount of tax = 1500 × 0.25
Amount of tax = $375
Therefore, Sally is required to pay a tax of $375
May I please get help with this I need help with finding the original Final point. I also need help with the graphing part, I am confused as to how I should do it after many tries Or where I should reflect it
Answers
We need to reflect the given figure across the y-axis. Then, we need to write the coordinates of the original and the reflected large dot.
When we reflect a point across the y-axis, its y-coordinate remains the same, and its x-coordinate changes the sign:
[tex](x,y)\rightarrow(-x,y)[/tex]The original coordinates of the large dot is: (5,3).
Thus, the coordinates after the reflection will be: (-5,3).
Doing the same with the coordinates of the other vertices and joining them, we obtain the reflected figure, as shown below:
question will be in picture
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Given that the total money Elise had, was $30.
Since she spent $18 on gifts and the rest on 3 corn dogs.
Let the cost of a corn god be 'x' dollars. Then the cost of 3 corn dogs will be,
[tex]\begin{gathered} \text{Expense on corn dogs}=\text{ No. of corn dogs}\times\text{ Cost of 1 corn dog} \\ \text{Expense on corn dogs}=3\times x \\ \text{Expense on corn dogs}=3x \end{gathered}[/tex]So it is found that Elise spent $(3x) on corn dogs.
Since she had spent all the money in buying these two things, the total exprense must be equal to the total money Elise had in the beginning,
[tex]\begin{gathered} \text{Expense on corn dogs}+\text{ Expense on gifts}=30 \\ 3x+18=30 \end{gathered}[/tex]Thus, the required equation is obtained.
Therefore, option (a) is the correct choice.
Over the past 6 seasons, one baseball player's batting averages were 0.248, 0.302, 0.248, 0.307, 0.295, and 0.369. A second player's batting averages were 0.349, 0.231, 0.272,0.263, 0.275, and 0.384. What are the range and mean of each player's batting averages? Use your results to compare the players' batting skills.Find the range and mean of the first player's batting averages.The range is (Type an integer or a decimal.)(Round to the nearest thousandth as needed.)The mean is
Answers
First let's write down the batting averages of the first player in ascending order:-
0.248, 0.248, 0.295, 0.302, 0,307, 0,369
The difference between the largest value and the minimum value will give us the range:-
So range for the first player will be = 0.369 - 0.248 = 0.121
Now let;s calculate the mean of the first palyer's batting averages
[tex]\begin{gathered} \text{Mean}_1=\frac{0.248+0.248+0.295+0.302+0.307+0.369}{6} \\ =\frac{1.769}{6} \\ =0.295\text{ (approx)} \end{gathered}[/tex]Now let's write down the batting averages of the second player in ascending order
0.231, 0.263, 0.272, 0.275, 0.349, 0.384
So the range for second player will be:-
0.384-0.231= 0.153
Now let;s calculate the mean of the second palyer's batting averages
[tex]\begin{gathered} \operatorname{mean}=\frac{0.231+0.263+0.272+0.275+0.349+0.384}{6} \\ =\frac{1.774}{6} \\ =0.296(approx) \end{gathered}[/tex]The mean for second player is 0.296 (approx)
And the mean for the first player is 0.295 (approx)
Since the mean avarage of both the players is almost same but the range of second player is more than that of first player, so the second player has good batting skills as compared to the first player.
The Elimination MethodTry These BSolve each system using elimination.a. 7x+5y=-14x - y=-16
Answers
Elimination Method : In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
The given system of equation:
7x + 5y = -1 (1)
4x - y = -16 (2)
Multiply the equation (2) by 5 :
5(4x -y ) = 5(-16)
20x - 5y = -80 (3)
Add equation (3) & (1)
(7x + 5y) + (20x -5y ) =(-1) + (-80)
7x + 5y + 20x - 5y = -1 - 80
27x = -81
x = -81/ 27
x = -3
Substitute the value of x =3 in the equation (1)
7x + 5y = -1
7(-3) + 5y = -1
-21 + 5y = -1
5y = -1 +21
5y = 20
y =20/5
y = 4
Thus, the solution of system is (x, y) = (-3, 4)
Answer : x =-3, y = 4
The regular price of an item is $350. The store is having a 25% off sale,plus an additional 20% off discount. What is the price, before tax, you would pay for this item?
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the price before taxes is:
[tex]350\cdot0.75\cdot0.8=210[/tex]$210 is the price before taxes
how to you write out and solve this math problem 1.79.1619.90
Answers
EXPLANATION
Considering the addition
Write the numbers one under the other, line up the decimal points.
Add trailing zeroes so the numbers have the same length.
1. 7 9
+ 0. 1 6
+ 1 9. 9 0
----------------
Add each columns of digits, starting on the right and working left.
If the sum of a column is more than ten, 'carry' digits to the net column on the left.
Add the digits of the bolded column: 9 + 6 + 0 = 15
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
Carry 1 to the column on the left and write 5 in the bolded column:
1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
5
Add the digits of the bolded column: 1 + 7 + 1 + 9 = 18
1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
8 5
Carry 1 on the colum to the left and write 8 in the bolded column:
1 1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
8 5
Place the decimal point in the answer directly below the decimal points in the term:
1 1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
.8 5
Add the digits of the bolded column : 1 + 1 + 0 + 9 = 11
1 1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
.8 5
Carry 1 to the column on the left and write 1 in the bolded column:
1 1 1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
1 .8 5
Add the digits of the bolded column: 1 + 0 + 0 + 1 = 2
1 1 1
0 1. 7 9
+ 0 0. 1 6
+ 1 9. 9 0
----------------
2 1 .8 5
The answer is 21.85
if angle 4 equals 140 what do the other angles equal to.
Answers
Find the equation for a polynomial f(x) that satisfies the following:Degree 5- Root of multiplicity 1 at 2 = 1- Root of multiplicity 2 at x = 2- Root of multiplicity 2 at x = -2y-intercept of (0,–32)
Answers
The equation for this polynomial is:
[tex]\begin{gathered} 2(x-1)(x-2)^2(x+2)^2 \\ 2x^5-2x^4-16x^3+16x^2+32x-32 \end{gathered}[/tex]So that's the equation we're asking for.
Both could be the answers. However, this is the final one:
[tex]2x^5-2x^4-16x^3+16x^2+32x-32[/tex]finding slop on the line
Answers
As per given diagram:
Take two points on the line (1,1) and (0,4)
For the slope of the line:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Now put the values in the formula:
[tex]\begin{gathered} m=\frac{4-1}{0-1} \\ m=\frac{3}{-1} \\ m=-3 \end{gathered}[/tex]So the slopw of the given line is -3.
1/8, 2/7, 1/2, 4/5 what are the next two numbers?
Answers
ANSWER:
5/4 and 2
STEP-BY-STEP EXPLANATION:
If we look closely, we notice that the pattern is that 1 is added to the numerator and one is subtracted from the denominator, as follows:
[tex]\begin{gathered} \frac{1}{8} \\ \frac{1+1}{8-1}=\frac{2}{7} \\ \frac{2+1}{7-1}=\frac{3}{6}=\frac{1}{2} \\ \frac{3+1}{6-1}=\frac{4}{5} \\ \text{therefore, the next two numbers are:} \\ \frac{4+1}{5-1}=\frac{5}{4} \\ \frac{5+1}{4-1}=\frac{6}{3}=2 \end{gathered}[/tex]What is the equation of the line perpendicular to the function f(x)=x^2+2x-2 at the point (1,1)?
Answers
Equation of a Line
The equation of a line that passes through the point (h, k) and has a slope m, is given by:
[tex]y-k=m(x-h)[/tex]This is known as the point-slope form of the line.
We already know the coordinates of the point (1, 1) but we don't know the value of the slope m. We will find it out by using the rest of the data.
Our line is perpendicular to the function:
[tex]f(x)=x^2+2x-2[/tex]At the given point. To find the slope of the tangent line, we use derivatives:
[tex]f^{\prime}(x)=2x+2[/tex]Substitute x = 1:
[tex]\begin{gathered} f^{\prime}(1)=2\cdot1+2 \\ f^{\prime}(1)=4 \end{gathered}[/tex]Now we know the slope of the tangent line, but our line is perpendicular to that line, so we find the perpendicular slope with the formula:
[tex]\begin{gathered} m_2=-\frac{1}{m} \\ m_2=-\frac{1}{4} \end{gathered}[/tex]We're ready to find the required equation. Substituting the coordinates of the point and the just-found slope:
[tex]y-1=-\frac{1}{4}(x-1)[/tex]This is the point-slope form, but maybe it's required to find the slope-intercept form. Multiply by 4:
[tex]\begin{gathered} 4y-4=-x+1 \\ \text{Add 4:} \\ 4y=-x+5 \\ \text{Divide by 4:} \\ y=-\frac{1}{4}x+\frac{5}{4} \end{gathered}[/tex]This is the answer is slope-intercept form
Where C(x) is in hundreds Of dollars. How many bicycles should the shop build to minimize the average cost per bicycle
Answers
The average cost is given by the next equation:
[tex]C(x)=0.1x^2-0.5x+5.582[/tex]The graph of the function is
As we can see the parabola opens upwards therefore the minim will be located in the vertex
[tex]x=\frac{-b}{2a}[/tex]in our case
b=-0.5
a=0.1
[tex]x=\frac{0.5}{2(0.1)}=2.5[/tex]2.5 hundred
Then
[tex]2.5\times100=250[/tex]Therefore, the shop should build 250 bicycles
An escalator at a shopping center is 200 ft long and has a vertical rise of 52 feet.What is the measure of the angle formed by the escalator and the ground? Round to the nearest degree
Answers
A right triangle is made, with measure:
From definition:
sin(α) = opposite/hypotenuse
From the picture: opposite to α is side of 52 long and the hypotenuse is 200 ft long. Then:
sin(α) = 52/200
sin(α) = 0.26
α = arcsin(0.26)
α = 15°
y = 43 - 9Complete the missing value in the solution to theequation.(3,
Answers
We have the following:
[tex]y=4x-9[/tex]We have a solution pair is (x, y), in this case then x = 3, replacing we have
[tex]\begin{gathered} y=4\cdot3-9 \\ y=12-9 \\ y=3 \end{gathered}[/tex]The answer is: (3, 3)
i need help woth this asap i cant get it wrong
Answers
The given equation is a quadratic equation. Recall, the standard form of a quadratic equation is expressed as
ax^2 + bx + c = 0
The given equation is
x^2 - 8x = - 128
By adding 128 to both sides of the equation, we have
x^2 - 8x + 128 = - 128 + 128
x^2 - 8x + 128 = 0
By comparing this equation with the standard form equation,
a = 1, b = - 8, c = 128
The formula for solving quadratic equations is expressed as
[tex]\begin{gathered} x\text{ = }\frac{-\text{ b }\pm\sqrt[]{b^2-4ac}}{2a} \\ By\text{ substituting the given values, it becomes} \\ x\text{ = }\frac{-\text{ - 8 }\pm\sqrt[]{-8^2-4(1\times128)}}{2\times1} \\ x\text{ = }\frac{8\pm\sqrt[]{64-512}}{2}\text{ = }\frac{8\pm\sqrt[]{-\text{ 448}}}{2}\text{ = }\frac{8\pm\sqrt[]{-64\text{ }\times\text{ 7}}}{2} \\ x\text{ }=\frac{8\pm(\sqrt[]{-64)}\times\sqrt[]{7}}{2} \\ \text{Note, }\sqrt[]{-\text{ 1}}\text{ = i} \\ \sqrt[]{-64}\text{ = 8i} \\ x\text{ = }\frac{8\pm8i\sqrt[]{7}}{2} \\ \text{Factoring out 2 in the numerator, we have} \\ x\text{ = }\frac{2(4\text{ }\pm4i\sqrt[]{7})}{2} \\ x\text{ = 4 }\pm4i\sqrt[]{7} \end{gathered}[/tex]Option A is correct
Find square root of 49 Find square root of 100
Answers
We are asked to determine the square root of 49, this is written mathematically as:
[tex]\sqrt[]{49}[/tex]This means that we need to determine a number that when multiplied twice yields 49, that is:
[tex]7\times7=49[/tex]Therefore:
[tex]\sqrt[]{49}=7[/tex]This is my math homework, I don’t understand how to find the seconds of the ball hit the ground
Answers
When the ball hits the ground, the height of the ball to the ground is h(t) = 0.
Therefore, we can now substitute h(t), and solve the function using quadratic formula.
[tex]\begin{gathered} h(t)=-16t^2+16t+400 \\ 0=-16t^2+16t+400 \\ \\ \text{The function is now in standard form where} \\ a=-16,b=16,c=400 \end{gathered}[/tex]Using the quadratic formula, substitute the following values a,b, and c.
[tex]\begin{gathered} t=\frac{ -b \pm\sqrt{b^2 - 4ac}}{ 2a } \\ t=\frac{ -16 \pm\sqrt{16^2 - 4(-16)(400)}}{ 2(-16) } \\ t=\frac{-16\pm\sqrt[]{256-(-25600)}}{-32} \\ t=\frac{ -16 \pm\sqrt{25856}}{ -32 } \\ t=\frac{ -16 \pm16\sqrt{101}\, }{ -32 } \\ \\ t=\frac{-16+16\sqrt{101}}{-32} \\ t=-4.52494 \\ \\ t=\frac{-16-16\sqrt[]{101}\, }{-32} \\ t=5.52494 \end{gathered}[/tex]We have two solutions, t = -4.52494, and t = 5.52494. However, we will disregard the negative time value.
Therefore, the ball will hit the ground after 5.52494 seconds.
Need to find equation f(x) = a(b)^xfor these 2 sets of points.(0, -3) (1, -³/2)
Answers
y = a b^x
Substitute the first set of points into the equation
-3 = a * b^0
-3 = a * (1)
-3 = a
y = (-3)* b^ x
Now using the second point
-3/2 = -3 * ( b)^1
-3/2 = -3 *b
Divide each side by -3
1/2 = b
y = -3 ( 1/2) ^x
Can someone please help me with this ? I just need the answer
Answers
SOLUTION:
The graph of g(x) is the graph of f(x) translated 5 units to the left.
Thus, the equation is;
[tex]g(x)=(x+5)^2[/tex]