In the circle, AB=42, BC=18, and CD=4. The diagram is not drawn to scale.
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The numbers of cookies in a shipment of bags are normally distributed, with a mean of 64 and a standard deviation of 4. What percent of bags of cookies will contain between 60 and 64 cookies?
We can standardize the values by using the formula:
z = (x - mu) / sigma
where:
x = value of interest (in this case, 60 and 64)
mu = population mean = 64
sigma = population standard deviation = 4
So, for x = 60,
z = (60 - 64) / 4 = -1
And for x = 64,
z = (64 - 64) / 4 = 0
Now, we can use a standard normal distribution table or calculator to find the area under the curve between z = -1 and z = 0. This represents the percentage of bags of cookies that will contain between 60 and 64 cookies.
Using a standard normal distribution table or calculator, we find that the area between z = -1 and z = 0 is approximately 0.3413 or 34.13%.
Therefore, approximately 34.13% of bags of cookies will contain between 60 and 64 cookies.
To solve this problem, we will use the properties of a circle.
Step 1: Identify the given information.
- AB = 42
- BC = 18
- CD = 4
Step 2: Understand the properties of a circle.
- In a circle, the line segment joining the center of the circle to a point on the circle is called the radius.
- The radius is always perpendicular to the tangent line at the point of contact.
Step 3: Apply the properties of a circle to the problem.
- In this case, we can infer that BC is a tangent to the circle at point B.
- This means that BC ⊥ AB (perpendicular to AB) at point B.
Step 4: Solve for the missing length.
- Since BC is perpendicular to AB, we can use the Pythagorean theorem to find the missing length.
- In a right triangle with sides a, b, and c, where c is the hypotenuse, the Pythagorean theorem states that a^2 + b^2 = c^2.
- In this case, we have a right triangle ABC, with AB as the hypotenuse and BC as one of the legs.
- Using the Pythagorean theorem, we can solve for the length of AB.
AB^2 = BC^2 + AC^2
Substituting the given values:
AB^2 = 18^2 + AC^2
AB^2 = 324 + AC^2
We know that AB = 42, so we can substitute that value as well:
42^2 = 324 + AC^2
1764 = 324 + AC^2
Now, rearrange the equation to solve for AC^2:
AC^2 = 1764 - 324
AC^2 = 1440
Taking the square root of both sides, we find:
AC ≈ √1440
AC ≈ 37.95
So, the length of AC is approximately 37.95 units.
Step 5: Answer the question.
- The length of AC is approximately 37.95 units.
To find the length of CD, we can use the property of tangents drawn from an external point to a circle.
In this case, the line segment AB is the tangent to the circle at point B, and the line segment CD is drawn from an external point C to the circle.
According to the property, the length of the tangent squared is equal to the product of the external line and the portion of the external line intercepted by the circle.
To calculate the length of CD, we can use this property:
AB^2 = BC * CD
Substituting the given values,
42^2 = 18 * CD
Simplifying,
1764 = 18 * CD
To isolate CD, divide both sides by 18:
CD = 1764 / 18
CD ≈ 98
Therefore, the length of CD is approximately 98 units.