Helpp! -- A piece of wire 5 inches long is to be cut in two pieces. One piece is x inches long and is to be bent into the shape of a square. The other piece is to be bent in the shape of a circle. Find an expression for the total area made up by the square and the circle as a function of x.
Answers
since, we know that for the circle, the perimeter is s+s+s+s or 4s, then 4s = x and so on.
now, for the circle, the perimeter is just the circumference, circumference of a circle is 2πr, thus 2πr = 5-x and so on.
anyway... so... the equation of the area made by those two in x-terms, is just their sum... so let's do that then
[tex]\bf \pi \left( \cfrac{5-x}{2\pi } \right)^2\quad +\quad \cfrac{x^2}{4^2}\\\\ -------------------------------\\\\ \pi\cdot \cfrac{(5-x)^2}{(2\pi )^2}+\cfrac{x^2}{4^2}\implies \cfrac{\pi (5-x)^2}{2^2\pi^2}+\cfrac{x^2}{4^2}\implies \cfrac{(5-x)^2}{4\pi}+\cfrac{x^2}{4^2} \\\\\\ \textit{let's use the LCD of }4^2\pi \textit{ to add them up} \\\\\\ \cfrac{4(5-x)^2+\pi x^2}{4^2\pi }\implies \cfrac{4(5-x)^2+\pi x^2}{16\pi }[/tex]
now, you can expand the squared binomial on the numerator, but there will not be any like-terms to simplify, so, it won't make much difference, so.... you can expand it or not.
The expression for the total area made up by the square and the circle as a function of x is f(x) = x^2/16 + (25 - 10x + x^2)/(4*Pi) square inches.
Explanation:To solve this problem, you need to know the formulas for the areas of a square (Area = side^2) and a circle (Area = pi * radius^2). If a piece of the wire with length x inches is to be bent into the shape of a square, then each side of the square will be x/4 inches long (since a square has 4 equal sides). Therefore, the area of the square = (x/4)^2 = x^2/16 sq.inches.
The remaining piece of the wire is 5-x inches long and is to be bent into the shape of a circle. The circumference of the circle is 5-x inches (since it uses the remaining wire), therefore the radius of the circle is (5-x)/(2*Pi) inches. Therefore, the area of the circle = Pi*((5-x)/(2*Pi))^2 = (25 - 10x + x^2)/(4*Pi) sq.inches.
The total area made up by the square and the circle is the sum of their areas, therefore the function representing the total area f(x) = x^2/16 + (25 - 10x + x^2)/(4*Pi) sq.inches.
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Related Questions
the length of the hypotenuse is:
6.
12.
36.
[tex]6 \sqrt{3[/tex]
Answers
1/2 = 6/x
1/2x=6
x = 6X 2 = 12
The length of the hypothenuse is 12
One school survey showed that 3 out of 5 students own a pet. Another survey showed that 6 out of 11 students owned a pet. Are these results equivalent? Explain your reasoning.
Answers
6/11 own a pet means that (0.545) or 54.5% own a pet
Conclusion, in the 1st class more people 60% own a pet whereas in the second class only 54.5% have a pet
A farmer is going to plant carrots on 5 3/14 acres, corn on 4 23/42 acres and peppers on 2 5/21 acres. If each acre requires 6 bags of fertilizer, how many bags of fertilizer does the farmer need to plant all the acres?
Answers
so
[tex]\bf 5\frac{3}{14}\implies \cfrac{5\cdot 14+3}{14}\implies \cfrac{73}{14} \\\\\\ 4\frac{23}{42}\implies \cfrac{4\cdot 42+23}{42}\implies \cfrac{191}{42} \\\\\\ 2\frac{5}{21}\implies \cfrac{2\cdot 21+5}{21}\implies \cfrac{47}{21}\\\\ -------------------------------\\\\ \cfrac{73}{14}+\cfrac{191}{42}+\cfrac{47}{21}\impliedby \textit{our LCD is just 42}\implies \cfrac{3\cdot 73+1\cdot 191+2\cdot 47}{42} \\\\\\ \cfrac{219+191+94}{42}\implies \cfrac{504}{42}\implies \cfrac{12}{1}\implies 12[/tex]
now, that's how many acres the farmer has in total
now, if each acre takes 6 bags of fertilizer, well, you surely know how much that is.
A figure has a vertex at (5, 2). If the figure has line symmetry about the y-axis, what are the coordinates of another vertex of the figure?
Answers
Answer:
[tex](-5,2)[/tex]
Step-by-step explanation:
We have been given that a figure has a vertex at (5,2). The figure has line symmetry about the y-axis.
We know that if a figure is symmetric about y-axis, then its x-coordinate changes to opposite sign and y-coordinate remains same.
We can see that x-coordinate of our given point is 5, so x-coordinate of another vertex of the figure would be -5.
Therefore, the point [tex](-5,2)[/tex] is symmetric about y-axis for our given point.
richerd works at an ice cream shop. regular cones get two scopes of ice cream and large cones get three scoops. One hot saturday richard scooped 234 regular cones and 156 large cones one scoop of ice cream is 3 ounces a tub of ice cream is 10 pounds how many tubs of ice cream did richerd use to make the cones?
Answers
234 x 2 = 468 scoops
156 x 3 = 468 scoops
468 + 468 = 936 total scoops
936 x 3 = 2808 ounces
16 ounces = 1 pound
2808/16 = 175.5 pounds
175.5/10 = 17.55 tubs
round answer as needed
Simplify the expression. 33 • 32 + 12 ÷ 4
Answers
The expression 33 • 32 + 12 ÷ 4 simplifies to 1059.
Explanation:To simplify the expression 33 • 32 + 12 ÷ 4, we follow the order of operations - performing multiplication and division before addition.
Multiply 33 and 32 to get 1056.Divide 12 by 4 to get 3.Now we can add the results: 1056 + 3 = 1059.
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A bird feeder is in the shape of a cylinder. It has a volume of about 100 cubic inches. It has a radius of 2 inches. What is the approximate height of the bird feeder? Use 3.14 for pi
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volume = pi x r^2x h
100 = 3.14 x 2^2 x h
100 =3.14 x 4 x h
100 = 12.56 x h
h = 100/12.56 = 7.9617
the height is approximately 8 inches tall
A cold front moved in last weekend. In eight hours overnight, the temperature outside dropped from 14 degrees to -10. What was the average temperature change for each hour ?
Answers
I really suck at math. please help =)
Answers
Suppose y varies directly with x, and y = 8 when x = –6. What direct variation equation relates x and y? What is the value of y when x = –2?
Answers
Answer:
Direct variation states that the relationship between two variables in which one is a constant multiple of the other one.
In other words, when one variable changes the other one changes in proportion to the first.
i.e, if y is directly proportional to x then, the equal will be of the form is, y= kx where k is the constant of variation.
Given: y varies directly with x, and y = 8 when x = –6
By definition of direct variation,
y = kx
Substitute the values of x = -6 and y=8 to solve for k;
8 = -6k
Divide both sides by -6 we get;
[tex]k = -\frac{8}{6} = -\frac{4}{3}[/tex]
Now, to find the value of y when x = 2 we have;
[tex]y = -\frac{4}{3}x[/tex]
Substitute the given value of x =-2 we have;
[tex]y = -\frac{4}{3} \cdot -2 = \frac{8}{3}[/tex]
Therefore, the direct variation related x and y is, [tex]y = -\frac{4}{3}x[/tex]
and the value of [tex]y =\frac{8}{3}[/tex] when x = -2
222+203 is rounded up to what?
Answers
[tex]Remember [/tex]↓
[tex]1-5[/tex] would rounded [tex]downward [/tex]
[tex]6-9[/tex] would be rounded [tex]upward [/tex]
The Kwon family has a rainwater catchment system they use to water their garden. After 3 days without rain, the depth of water in the tank is 63 inches. After 5 days, the depth is 57 inches. What will the depth of water in the tank be after 17 days?
Answers
if d = 3
f(d) = 72 -3d = 72 - 3(3) = 72 - 9 = 63 inches
if d = 5
f(d) = 72 -3d = 72 - 3(5) = 72 - 15 = 57 inches
so
if d = 17
f(d) = 72 -3d = 72 - 3(17) = 72 - 51 = 21 inches
answer
the depth of water in the tank will be 21 inches after 17 days
63-57 = 6 inches in 2 days
6/2 = 3 inches per day
17-5 = 12
12*3 =36 inches in 12 days
57-36 = 21 inches after 17 days
Round to the nearest while decimal 6.7
Answers
Then, [tex]six [/tex] is rounded [tex]1ten[/tex] & [tex]seven[/tex] is rounded to [tex]ten [/tex]
Your answer: [tex]7[/tex]
[tex]good\\luck \\ on \\ your \\ assignment \\ \\ \\ \\ enjoy \\ your \\ day \\ \\ \\ \\ \\ \\ MeIsKaitlyn :)[/tex]
A total of 504 tickets were sold for the school play. They were either adult tickets or student tickets. There were 54 more student tickets sold than adult tickets. How many adult tickets were sold?
Answers
s = a + 54
a + (a + 54) = 504
2a + 54 = 504
2a = 504 - 54
2a = 450
a = 450/2
a = 225 <=== adults
a + s = 504
225 + s = 504
s = 504 - 225
s = 279 <=== students
Using algebra, we find that 225 adult tickets were sold out of a total of 504 tickets.
Explanation:Let's denote the number of adult tickets as x. Since there were 54 more student tickets sold than adult tickets, we can represent the number of student tickets as x + 54. The total number of tickets sold is 504, so we set up the equation x + (x + 54) = 504.
Combining like terms, we get 2x + 54 = 504. Subtracting 54 from both sides gives us 2x = 450. Finally, dividing both sides by 2 gives us x = 225.
Therefore, 225 adult tickets were sold.
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In how many different ways can you make exactly 0.75 using only nickles dimes and quarters if you have at least one of each coin?
Answers
A marina is in the shape of a coordinate grid. Boat A is docked at (4.2, −2) and Boat B is docked at (−5.2, −2). The boats are ____ units apart. A) 6.2 B) 7.2 C) 9.4 D) 13.4
Answers
d^2=(x2-x1)^2+(y2-y1)^2
d^2=(4.2--5.2)^2+(-2--2)^2
d^2=(9.4)^2
d=9.4
Answer:
Your answer will be D.
Step-by-step explanation:
I did it on the test and got it correct.
Regroup to express the number in a different way , please help
Answers
hope it helps
How many millimeters are there in 5 meters?
Answers
1 meter = 1000 mm
so 5 meters = 5 x 1000 = 5,000 millimeters
What is the limit for lim x -> 2 int x
Answers
2^2 / 2 equal to 2. Hence the answer to this problem is equal to 2.
A total of 487 tickets were sold for the school play. They were either adult tickets or student tickets. There were 63 fewer student tickets sold than adult tickets. How many adult tickets were sold?
Answers
s+a=487, using s from above in this equation you get:
a-63+a=487 combining like terms
2a-63=487 adding 63 to both sides
2a=550 dividing both sides by 2
a=275
So there were 275 adult tickets sold.
Final answer:
To determine the number of adult tickets sold for the school play, a system of equations is set up and solved, revealing that 275 adult tickets were sold.
Explanation:
To find the number of adult tickets sold for the school play, we need to set up a system of equations based on the information given. Let x represent the number of adult tickets and y represent the number of student tickets. According to the problem, the following two statements are true:
The total number of tickets sold is 487: x + y = 487
There were 63 fewer student tickets sold than adult tickets: y = x - 63
We can substitute the second equation into the first to find the value of x:
x + (x - 63) = 487
2x - 63 = 487
2x = 487 + 63
2x = 550
x = 275
Therefore, 275 adult tickets were sold.
Find the value of y, rounded to the nearest tenth. Please help me, I'd appreciate it!!
Answers
y² = 7(15+7)
y² = 7*22
y² = 154
y = √154
y = 12.4 ← to the nearest tenth
True or False: The sample size you need to estimate the population distribution should always be at least 10% of the population size.
Answers
False. The sample size needed to estimate the population distribution should not always be at least 10% of the population size.
Explanation:False. The statement that the sample size needed to estimate the population distribution should always be at least 10% of the population size is not true. The size of the sample needed depends on various factors such as the original population, the level of confidence desired, and the margin of error allowed. For example, if the population is large and diverse, a smaller sample may still provide a reliable estimate of the population distribution. It is important to consider statistical concepts such as normal distribution and sampling techniques when determining the appropriate sample size.
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the base of this solid crate has an area of 6 square centimeters the height of the crate is 4 meters what is the volume of the crate
Answers
The length * width is the area of the base, so:
[tex]Area Of Cuboid = base*height = 6 * 4 = 24 meters^3[/tex]
Answer:
2 × 10³ cm³
Step-by-step explanation:
Given data
Area of the base: 6 cm²Height of the crate: 4 m = 4 × 10² cmConsidering the crate is a cuboid, its volume (V) is:
V = length × width × height
Since
area of the base = length × width
We get
V = area of the base × height
V = 6 cm² × 4 × 10² cm
V = 2.4 × 10³ cm³ ≈ 2 × 10³ cm³ (we round off to 1 significant figure)
Evaluate the limit, if it exists. (if an answer does not exist, enter dne.)lim h → 0 (x + h)3 − x3h
Answers
[tex]\displaystyle\lim_{h\to0}\frac{(x+h)^3-x^3}h[/tex]
Now, if you're familiar with the definition of the derivatives, you'll notice that this is the limit form of the derivative of the function [tex]f(x)=x^3[/tex], which you may also know to be [tex]3x^2[/tex]. But let's assume you don't know that just yet, and that it's actually the result you intend to find.
Expand the numerator:
[tex]\dfrac{(x+h)^3-x^3}h=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}h=\dfrac{3x^2h+3xh+h^3}h[/tex]
Now, when [tex]h\neq0[/tex], we can divide through by the lowest power of [tex]h[/tex]. We can do this because we're considering the limit as [tex]h[/tex] is *approaching* 0, and not when it actually takes on the value of [tex]h=0[/tex].
[tex]\dfrac{(x+h)^3-x^3}h=3x^2+3xh+h^2[/tex]
Now, as [tex]h\to0[/tex], we can see only the leading term remains, so that
[tex]\displaystyle\lim_{h\to0}\frac{(x+h)^3-x^3}h=3x^2[/tex]
as expected.
To evaluate the limit of (x + h)³ - x³h as h approaches 0, we can expand the expression using the binomial theorem and simplify. The resulting expression can be factored to show that as h approaches 0, the entire expression becomes 0. Therefore, the limit is 0.
Explanation:To evaluate the given limit, we can start by expanding the expression (x + h)³ using the binomial theorem. This gives us (x³ + 3x²h + 3xh² + h³) - x³h. Simplifying further, we can cancel out the x³ terms and obtain 3x²h + 3xh² + h³ - x³h. Now, we can factor out h from the expression to get h(3x² + 3xh + h² - x³). As h approaches 0, the entire expression becomes 0, since h is being multiplied by a polynomial that does not contain h. Therefore, the limit is 0.
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Linda deposits $500 into an account that pays simple interest at a rate of 4% per year. How much interest will she be paid in the first 4 years?
Answers
Year 1 - $500 * 1.04 = $520
Year 2 - $520 * 1.04 = $540.80
Year 3 - $540.80 * 1.04 = $562.432
Year 4 - $562.432 * 1.04 = $584.92928
Aftere four years. Linda would be paid $84.93
Express the volume of a cone, V, as a function of its radius, r, if the radius is 1/5 of the height.
Answers
Sara must plant 340 trees. In the past 6 days Sara planted 204 trees. If she continues at this rate, how many more days will it take her to plant all the trees?
Answers
The first card selected from a standard 52-card deck was a king. if it is not returned to the deck, what is the probability that a king will be drawn on the second selection?
Answers
there would be 3 kings left and 51 cards left
so it would be a 3/51 which reduces to 1/17 probability
The probability of drawing a king from a 52-card deck on the second draw, given that a king was drawn on the first draw and was not replaced, is 1 in 17.
Explanation:The question pertains to the concept of probability, specifically with regards to sampling without replacement in a 52-card deck. First, let's identify the elements: there are 4 kings in a 52-card deck. When one king is drawn and not replaced, there are now 51 cards left with 3 kings.
The probability of drawing a king on the second draw, with the first king not replaced, is the number of favorable outcomes (drawing a king) divided by the total number of outcomes (total cards left). Hence, the probability is 3/51 = 1/17.
This means that there is 1 chance in 17 of drawing a king on the second draw if a king has been drawn on the first draw and not replaced.
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Sales tax on an item is directly proportional to the cost of the item purchased. If the tax on a $500 item is $30, what is the sales tax on a $900 item?
Answers
cross multiply
(500)(x) = (900)(30)
500x = 27000
x = 27000/500
x = 54 <===
Answer: $54
Step-by-step explanation:
The equation to show the direct variation in two quantities x and y is given by :-
[tex]\dfrac{x_1}{y_1}=\dfrac{x_2}{y_2}[/tex]
The tax on a $500 item is $30.
Let 'x' be the sales tax on a $900 item.
Then , we have the following equation:-
[tex]\dfrac{x}{900}=\dfrac{30}{500}\\\\\Rightarrow\ x=\dfrac{900\times30}{500}\\\\\Rightarrow\ x=54[/tex]
Hence, the sales tax on a $900 item = $54
Three counters are used for a board game.If the counters are tossed,how many ways can at least one counter with Side A occur?
Answers
Probability = n!/r!(n-r)! * p^(n-r) * q^r
The p is the probability of getting a side A in one toss. Since a counter has only two side, p = 0.5. The q is the probability of not getting side A in one toss, which is also q = 0.5. Now, r is the number of success per n trials. There are 3 tosses so, n=3. The question is getting "at least 1" counter. So, r=1, r=2 and r=3.
Probability for r=1: 3!/1!(3-1)! * (0.5)^(3-1) * (0.5)^1= 0.375
Probability for r=2: 3!/2!(3-2)! * (0.5)^(3-2) * (0.5)^2= 0.375
Probability for r=1: 3!/3!(3-3)! * (0.5)^(3-3) * (0.5)^3= 0.125
Total probability = 0.375 + 0.375 + 0.125 = 0.875